Just like for the 1st round, I'll put this in few categories. (also note that this is just my opinion):
Character design and personality
Misty - I love both of her designs. IIRC she is only Gym Leader that had huge design change. Her design fits her Gym and theme.
Bugsy - Just like with Falkner, I think Bugsy's design really fits the whole game, even though he is just "Bug Catcher with purple hair".
This one is quite hard, because I like both of their designs... But I'll give this one to Misty, because of her LGPE re-design.
Pokémon
Let's talk about Pokémon team when you face them:
Misty - In RBY/FRLG she uses Staryu and Starmie. They can prove hard, especially in remakes, because her Pokémon knowing Water Pulse and Recover. Once again, it's hard for the Charmander users, but you can catch Pikachu, if you're having a hard time. Overall I think her battle is challenging enough for the 2nd Gym. In LGPE she uses Psyduck and Starmie with reduced levels, for some reason, making the battle ridiculously easy.
Bugsy - Now Bugsy's team is interesting. In GSC, his Pokémon even have better movesets for some reason and Scyther's Fury Cutter can prove to be hard. But his Pokémon's levels are just too low for 2nd Gym. HGSS reworks movesets and adds 1 level to every Pokémon. Scyther no longer knows Furry Cutter, but knows U-turn. It changes the pace of the battle, but isn't that bad. His other 2 Pokémon are Metapod and Kakuna, which imho aren't that great for a Gym Battle.
Once again, Misty's team is more appealing.
Gym design and grunts
This one will be really short. Both Gyms are simple and fit the theme. But I like the puzzle in Bugsy's Gym in HGSS, so I'll give this one to him.
Actual battle
Now, let's look at the battle using the teams above (Lapras' sprite is only one that is animated btw and it drives me crazy). As others already said, there is no way Misty can beat Bugsy's Shedinja. None of her Pokémon knows Flying, Rock, Ghost, Fire or Dark-Type attack to take it down. It's clearly lost battle from beginning. No doubts Bugsy won this battle... Or is there a way? It took me months of research, but I finally found out way how can Misty beat that pesky immortal bug. And it was in front of me all this time... That moving Lapras. She knows Perish Song! But to use it successfully, Shedinja has to be Bugsy's last Pokémon and Misty will team another 1 or 2 Pokémon to secure her victory. So, is it possible for this scenario to happen? I would say there is. Bugsy's Pokémon's levels are in range 48 (Shedinja) - 56 (Scizor), while Misty's are 54 (Floatzel and Lanturn) - 60 (Milotic and Starmie). That give Misty a small advantage, but almost none of her Pokémon are super effective against Bugsy's. On the other side, Bugsy has several Pokémon effective against Misty's.
Okay, enough talk, let's battle! Bugsy's Yanmega can fall against Starmie, Floatzel, Milotic, Lapras and Lanturn. That's all you got from Misty's side for the super effective hits. Bugsy's Scizor and Heracross easily cut through Misty's team, but I would say together they would take down 3-4 Pokémon - Starmie, Floatzel, Lanturn and maybe Quagsire. That would leave both Gym Leaders with 3 Pokémon. Bugsy's Pinsir can be easily countered with Quagsire, assuming she survived earlier and Guillotine won't hit. Vespiquen could be annoying, but she is at level 52, so no real problem. She could probably finish of Quagsire, though. That would leave Bugsy with only Shedinja and Misty with Milotic and Lapras. Plus Milotic has Rest, so she can heal pretty well.
So yeah, there is a way for Misty to win this fight. But that's just 1 way in all the possible ways for this battle, so I'll give Bugsy this one.
Well, this was pretty nice matchup. I'll vote Misty, because she has bigger impact in the games that Bugsy.