I have tried the following ones (Note: all GOG versions and all in regular Wine, but I suppose it is mostly the same in Proton):

• Harry Potter 1-4: Runs great. One of the levels is bugged, however (also on Windows), so I had to mess with some dynamic light settings.
• Harry Potter 5-7: Runs okay. The sound was severely bugged when I played it (no sound attenuation, so everything always had maximum volume - including random swarms of flies). This might have been fixed with FAudio.
• Batman 1: Runs great, didn't have any problems (except for a few random crashes, which also seem to happen on Windows).
• Batman 2: Screen freezes when first showing a cutscene when starting a new game. Sound continues for a few minutes and the game then crashes. I haven't found a way to make it work.

Also, please note that many of the LEGO games are quite buggy in general, so not all of these issued may be caused by Wine/Proton.

And do remember you can support GOL using are affiliate address ;)

https://www.humblebundle.com/games/lego-games-bundle?partner=gamingonlinux

Same as Anarus here. I only own the Lego games from GOG.com and have only tried them via wine-staging.

Lego Harry Potter 1-4 works flawlessly, I never had any problems, controller support included.
Lego Batman works, but I never got a controller working, I have to play the game with keyboard.

And in case they are going to add some more games, Lego Star Wars Complete Edition works fine as well, albeit controller support is not so great there either. Lego Pirates of the Caribbean again works flawlessly, controller support included.