I played WoW through Lutris (and later Bottles) with minimal issues for all of classic TBC and WotLK. Basically use either platform to install and run the Battle.net client, and then use bnet to install any blizz apps like normal. I used WowUp to manage addons. WoW should not be a blocker for you.
That said, I'm thoroughly done with blizzard's shit and won't be playing wow anymore.
IMO bazzite is too focused on gaming for people to be daily driving it for everything, but hey whatever works. Just hope they're not upset when something breaks and the response from bazzite is "well yeah, that's not something we bother testing for".
(I have bazzite on a HTPC in my living room, and I think it's perfectly suited for that usecase)
IMO Mint, Fedora, or OpenSUSE is going to offer the more stable, user-friendly experience long term. Install Lutris through the distro's package manager, launch it, install bnet through lutris, launch it, install wow through bnet, launch it, Thrall's your uncle 😉.
Edit: to answer your other question, yes Lutris runs as an app similar to how battle.net or steam works on windows. It's just that instead of having a storefront and downloading data directly from a central "lutris" server, it's basically a bunch of community-written scripts to automate the installation and configuration of games from all sorts of places. So when you tell lutris to install bnet, it's running a script that goes and downloads it from blizzard, then locally creates a wine environment, launches the installer in that environment, you install it like on windows, and then it creates a lutris launcher entry for the bnet executable so that when you click play on it in lutris, it will automatically launch it in a wine environment each time.
And it should all work in KDE plasma, gnome, cinnamon, or whatever window manager you're using (the window manager on msft windows is called dwm and it's responsible for the same job).