I first downloaded and played it on my main rig(also linux). I used the lutris community script to install battle.net and then just installed the game through the launcher as you normally would. Then I simply copied the prefix to my SD, that way I didn't have to redownload it again, added it to lutris on my SD, linked it to steam, and voila. I'm using Proton-GE 8.4 and have had no issues so far.
You could just do it all on the SD directly I guess.
Monster Hunter World and/or Rise.