I'm largely going to ignore the Rival as they are not difficult past Rival 2. I picked Mudkip as my starter because Grovyle is the easiest Pokemon of the starters to defeat.
Roxanne
Team: Zigzagoon, Marill
Zigzagoon is useless here.
Marill can take out the Geodudes with Water Gun. To defeat Nosepass, Marill will need to use Defense Curl a few times and then Water Gun until it dies, chugging some Potions as needed. You can use the X Special near Rusturf Tunnel to make this go faster.
Brawly
Team: Zigzagoon, Azumarill (evolved on the Gym Trainers)
Zigzagoon can handle Meditite but is otherwise useless.
Azumarill wins by using Defense Curl on Machop and then using Rollout. 3 hits killed Machop, the 4th hit kills Meditite, and the 5th hit kills Makuhita.
Rival 2
Team: Linoone, Azumarill, Gloom
Linoone just spams Headbutt. Combusken or Marshtomp may give it issues.
Gloom doesn't like Slugma or Combusken but otherwise wins without much of a problem.
Azumarill just uses Rollout. I'm not sure if this works against Marshtomp, but it steamrolls Grovyle.
Wattson
Team: Linoone, Azumarill, Gloom
Linoone struggles to handle Magneton but does alright against the others with Mud Sport helping it tank hits. It can in theory beat Magneton with Swagger (Slateport tutor) + Rock Smash but I never got lucky enough for this to pan out.
Azumarill is bad here, STAB Shock Wave does too much damage to get Rollout going. It is walled by Magneton and outsped by everything.
Gloom hates Magneton but it handles the others fine with Bullet Seed and powders. You might want an X Special to do more damage.
(Wattson is actually pretty hard without a Ground-type or Combusken/Hariyama, who knew)
Maxie 1
Team: Linoone, Azumarill, Gloom, Seviper, Skarmory
Linoone can sweep if Intimidate is removed, it takes out Mightyena 1v1, beats Zubat easily, and can Dig through Camerupt if it decides not to Magnitude.
Azumarill Bubblebeams Camerupt and takes out the rest with Strength. May need a Super Potion or two.
Gloom is not great here but it can beat Mightyena at least. Seviper is similar, though it is less bad against Zubat.
Skarmory does not like Ember from Camerupt but handles everything else with little issues.
Flannery
Team: Linoone, Azumarill, Gloom, Seviper, Skarmory
Skarmory is useless here.
Gloom is also useless here, but it can maybe take out Slugma or Numel with Sleep Powder luck.
Seviper can handle Slugma but loses to everything else.
Linoone can take out Slugma and Numel with Return and do a solid chunk to Camerupt before Overheat murders it.
Azumarill solos with the Defense Curl + Rollout combo.
Norman
Team: Linoone, Azumarill, Gloom, Seviper, Skarmory
Linoone can solo this fight if played correctly. Give it a Persim Berry and use an X Attack on turn 1. The berry is to absorb Spinda's Teeter Dance. +1 Return should 2HKO everything but Slaking. Against Slaking you want to Dig into Return range (which should be outside Hyper Potion range) and then Return to KO. Dig allows you to dodge Slaking's attacks so you don't get bodied by Facade or Counter.
Azumarill can potentially win this fight with Defense Curl + Rollout. You will want to hold a Persim Berry for Teeter Dance and you may need an X Attack to KO Slaking. If Slaking survives a hit you have a good chance of getting iced by Counter.
Do not use Gloom here. It does little damage and attempts to status with powders are punished by Facade.
Seviper is too slow and frail to do much aside from getting a stray hit on Vigoroth.
Skarmory solos this fight with ease as long as you give it the Toxic TM so it can kill Slaking without triggering Counter. It handles everything else by being Skarmory.
Winona
Team: Linoone, Azumarill, Gloom, Seviper, Skarmory, Xatu
Linoone easily soloes with Thunderbolt and Ice Beam.
Azumarill soloes with Defense Curl + Rollout.
Gloom sucks here.
Seviper can actually almost solo this fight, but Altaria does beat it with Earthquake. It handles everything else though.
Skarmory soloes this fight by being Skarmory. I don't recommend fighting her Skarm with your Skarm though, unless you have Rock Smash to debuff her Defense.
Xatu almost solos this fight. It can beat Skarmory (3HKO with Night Shade) but isn't strong enough to beat Altaria with Psychic before Altaria beats it.
Maxie 2 (Magma Hideout)
Team: Linoone, Azumarill, Gloom, Seviper, Skarmory, Xatu
Linoone does decent damage and can potentially solo if you are able to absorb a Swagger with a Persim Berry, but it has issues against confusion otherwise and Camerupt's Earthquake wrecks it.
Azumarill can get trolled by Confusion but otherwise does fine, may need a potion.
Gloom is bad here.
Seviper gets trolled by confusion, can have some issues killing Crobat, and gets bodied by Camerupt.
The only threat to Skarmory here is confusion due to Camerupt having no Fire attacks.
Xatu handles Crobat and can chunk Camerupt for good damage before Rock Slide claims it.
Tate and Liza:
Team: Linoone, Azumarill, Bellossom, Seviper, Skarmory, Xatu (now complete)
There are many combinations here so I will just go over individual performances.
Linoone: It outspeeds and 2HKOes Xatu with Shadow Ball. Does good damage to everything else with SBall, but can't take more than one hit.
Azumarill is actually bad here. It's slow and does little damage with its weak Surfs, and STAB Psychics will quickly ruin its day since when its on the team it has a tendency to get ganged up on.
Bellossom sets up Sunny Day and OHKOes everything but Xatu with Solar Beam. As long as you have a partner to handle Xatu, it wins easily. Linoone + Bellossom ended up being the ideal pair for my team to handle this fight.
Seviper is outsped and OHKOed by everything. It is good as a distraction but nothing else.
Skarmory can't handle Solrock's Flamethrower, but takes everything else and does good damage to the rocks.
Xatu takes the hits well, but Giga Drain is pretty weak here. I can definitely see it doing better with Sunnybeam, but Bellossom needed it more. Giving it Shadow Ball might help too.
Maxie 3 (Space Center):
If its good in Maxie 2, its good here too. Bellossom still hates the bats but otherwise does way better here thanks to Solarbeam being a far better move than Bullet Seed.
Archie:
Maxie 2 MUs largely apply here.
Juan:
(not listing the team anymore because it is final)
Linoone can eat an X Attack and then KO Luvdisc with Return. You probably won't sweep without additional setup, but you can probably take out another mon or two before going down.
Azumarill can kill things with Return/Brick Break and some potions. It can struggle against Whiscash (Earthquake hurts) and Kingdra probably beats it with Double Team and Rest.
Bellossom solos easily with Sunnybeam. Solarbeam is a clean 2HKO on Kingdra.
Seviper can take out Luvdisc and maybe Sealeo. Bad vs the others.
Skarmory doesn't like the Water moves but can set up Spikes to make killing Kingdra easier.
Xatu solos by setting up CM on Luvdisc. Giga Drain kills Crawdaunt and Psychic handles everything else.
E4 Movesets:
bellossom: magical leaf/solarbeam/sunny day/sleep powder
seviper: sludge bomb/crunch/flamethrower/glare
xatu: psychic/fly/giga drain/calm mind
linoone: return/shadow ball/ice beam/thunderbolt
azumarill: return/surf/rollout/brick break (bb replaced by blizzard for drake)
skarmory: steel wing/fly/toxic/spikes
Sidney:
Linoone struggles to do much here without a lot setup. It can probably take out one mon, but no more than that.
Azumarill loses to Cacturne (it outspeeds and Needle Arms you) but otherwise handles this fight fine with Brick Break and some potions.
Bellossom does not like dealing with Shiftry's Double Team + Swagger shenanigans, but can otherwise solo this fight with the raw power of Sunnybeam against his relatively frail team.
Seviper can handle Cacturne and possibly Mightyena. Shiftry beats it with Extrasensory and everything else beats it in a straight fight.
Skarmory doesn't like Crawdaunt's Surf, but it handles everything else with ease.
Xatu is bad here for hopefully obvious reasons.
Phoebe:
Linoone can take out the Banettes and beat Dusclops 1 with an X Attack. Curse and Sableye's Double Team makes soloing with Linoone impractical.
Azumarill sucks in this fight due to having to rely on weak Surfs for damage.
Bellossom can solo this fight with 1 X Special (to 2HKO the Level 51 Dusclops). Sleep Powder helps setup, as long as Dusclops 1 does not Curse you Bellossom should win handily.
Seviper can actually solo this fight if some specific circumstances are met; namely, it needs 2 X Specials to 2HKO Dusclops 2 with Crunch (it takes Earthquake at full HP) and it needs a +Speed nature to outrun the Banettes. Seviper also needs to not get Cursed by Dusclops 1 while setting up. If Seviper is outsped by the Banettes, it loses because it can't take enough hits to sweep.
Skarmory has a hard time doing much here, it can get trolled by Curse/Wisp and one of the Banettes has Thunderbolt. It can set up Spikes on Dusclops 1 to help its teammates out though.
Xatu has no safe setup and is walled by Sableye.
Glacia:
Linoone can probably handle one of the Sealeos. It takes too much damage from the Glalies and the Walrein to beat them.
Azumarill can mostly handle this fight with Brick Break and some potions. You may want an X Attack to guarantee 2HKOes, otherwise you'll have to go for some damage rolls. There are two threats to Azumarill here: the L52 Glalie OHKOes with Explosion and is faster, and Walrein might ice Azumarill with Sheer Cold before it can take it down.
Bellossom can solo this fight in Emerald - it puts the L50 Sealeo to sleep with Sleep Powder, eats an X Special, sets up Sunny Day, and then sweeps with Solarbeam. You need Sealeo to sleep 3 turns to pull this off, because you will run out of sun on the second Glalie and you need to be at full HP to take an Ice Beam from it to put the sun back up. The X Special is needed to secure the OHKO on Walrein with Solarbeam. +1 Solarbeam is a damage range to OHKO the Glalies. This is kind of risky and relies on getting enough Sleep turns but it can work. Vileplume is probably more reliable at this due to its better SpA. Probably not workable in RS because she leads Glalie, but if you can force in one of the Sealeos you can try to pull off a midbattle setup.
Seviper chips something with Sludge Bomb or Flamethrower and then dies.
Skarmory takes too much damage from Ice Beam to do much other than lay Spikes.
Xatu can actually set up on the L50 Sealeo because it likes to use Body Slam instead of Ice Ball against it. You need all 6 boosts to KO Walrein though, but it can be done. Everything else kills it.
Drake:
I could not solo Drake with any member of my team so I will just discuss how the battle went and then make some additional notes. Before this fight I dropped Brick Break on Azumarill for Blizzard so it could contribute.
Skarmory leads vs Shelgon. Skarmory beats Shelgon 1v1 with Steel Wing and Toxic. Flygon comes out, switch to Azumarill on Flamethrower. Azumarill takes an Earthquake and Blizzards Flygon for about 70% damage. Switch to Xatu on EQ, Xatu takes a Crunch finishes off Flygon. Salamence comes out and takes out Xatu with Rock Slide. Seviper comes in, takes a Dragon Claw from Salamence (barely) and paralyzes it with Glare. Sevipers chips about 25% HP away with Sludge Bomb before it dies. Linoone comes in and 2HKOes with Ice Beam through Sitrus, barely living a Dragon Claw. Altaria comes out, takes an Ice Beam from Linoone for about half before Linoone dies. Skarmory comes back in and finishes off Altaria. Kingdra KOes Skarmory. Bellossom comes out and defeats Kingdra by putting it to sleep, then setting up Sunny Day and 2HKOing with Solarbeam. GG.
Notes: Skarmory does better in RS because the Kingdra is replaced by a Fireless Flygon. Xatu has no safe setup targets. This was probably the one time Seviper was useful in the entire E4.
Wallace:
Similar boat to Drake:
Bellossom leads vs Wailord. Wailord uses Rain Dance which is immediately countered by Bellossom using Sunny Day. Bellossom OHKOes Wailord with Solarbeam. Milotic (somewhat bafflingly) comes out before Tentacruel. Milotic is put to sleep and 2HKOed by Solarbeam through Sitrus. Tentacruel comes out as sun expires. Seviper is sacrificed, Azumarill comes out. Azumarill takes Tentacruel to red with Return before succumbing to Toxic and Sludge Bomb. Xatu comes out and Calm Minds on the Full Restore turn. Xatu 2HKOes Tentacruel with +1 Psychic and takes a Sludge Bomb. Ludicolo comes out and is chipped with Psychic once for about 33% before Xatu goes down to Surf. Linoone comes out and finishes off Ludicolo with two Returns, taking a Surf in the process. Whiscash comes out, Bellossom comes in and eats up an Earthquake, taking a second as it uses Sunny Day. Bellossom OHKOes Whiscash with Solarbeam. Gyarados comes out, it takes about half from Solarbeam and kills Bellossom with Hyper Beam. Linoone comes in and finishes off Gyarados with Thunderbolt. GG.
Notes: Xatu's only safe setup target is Whiscash. Gyarados was able to take +6 Psychic from it and OHKOed Xatu with Hyper Beam, so Xatu setup probably isn't reliable here. Not having an Electric move on your CMing Psychic here bites. Bellossom can solo this fight but it needs a lot of setup (3 X Specials is required to OHKO Tentacruel with Solarbeam), so unless you feel like rolling the dice on sleep turns its easier to just let a teammate handle Tentacruel. Seviper can maybe take out Ludicolo with some luck. Skarm is pretty bad here but probably does better in RS.