(Note: For most of this post, I'm leaving the dot out of Battle.net. This is because the forum autocensor once BANNED a user in a thread about Battle net because the name is also a URL, and it decided the user was spamming. It even auto deleted replies. Mods were able to recover posts, but not the user account, so I'm taking no chances.)
I'm not familiar with Faugus, but I wrote up a walkthrough for getting Battle.net working via Bottles and Heroic some time ago. The walkthroughs are a bit older, but a user relatively recently said my Heroic instructions worked perfectly for them the first try. You can find that here:
[HOW TO] Use Battle.Net on Zorin (Diablo 3, Diablo 4)
To more directly speak to your issue, Steam may well give you better performance than Faugus, but it depends entirely on what was ruining Faugus' performance. If it's your video driver, Steam will likely turn out exactly the same. If it's related to Proton (the software that makes Windows games run under Linux in most cases), Steam might do much better.
Running Battle net under Steam is entirely possible. In a nutshell, Battle net's installer needs to be added to Steam as a non-Steam game. This is done by downloading the installer, choosing Games > Add a Non-Steam Game in Steam, and then following the prompts. Once you're done, you'll have the Battle net installer (not Battle net itself) in your Steam library list. Select it and click play as if it were a game, and you'll go through the Battle net install process. Now you'll have to repeat the Add a Non-Steam Game steps again, but adding Battle net itself. This will require finding it, and to be honest, I'm not sure where it will be, as I've done this with other launchers, but not recently. Once Battle net itself is in your library list, you can run that, and install whichever games you want to install. You can also remove the Battle net installer from your library, since you'll be using the Battle net entry. Sorry these instructions are not more precise--I wrote two detailed walkthroughs linked above for other methods, but I don't use Steam for Battle net personally.
One important caveat: Blizzard is known for breaking Battle net in Linux from time to time. It's not deliberate. They're making changes that break Linux because they don't CARE about Linux, and then the Proton developers have to clean up and make it work again. Accordingly, you may need to change the version of Proton you use with Battle net via Steam. There are instructions on how to do that in this forum as well.
If you require more detailed steps for getting Steam to work, I can try to provide them this weekend, but please give what's here a shot, just in case it's enough for you.