If the stuff I’ve used has an app on windows, I don’t know about it. All my audio gear has been plug and play in Linux without issues. I’m going to guess that if you have Bluetooth working in general on your device, the headphones won’t be a problem themselves.

Based on what you are looking for, the only thing I can recommend specifically is ModMics from Antlion Audio - they specialize in good¹ mics as an addon for a good set of headphones, I believe using an adhesive pad to attach it where you want it. Basically, instead of paying extra for lower quality stuff “for gamers”, find a good pair of headphones from a reputable audio brand, and then slap a mic on the side.

For reference - I gave up on “gaming” options a while ago. I have a Scarlet Solo audio interface with an AudioTechnica XLR mic, AudioTechnica ATH-AD700X headphones, and then Edifier studio monitors for speakers with a Mackie Subwoofer. It all runs off the Scarlet box though, so that is the only thing plugged directed to the PC. I do still have a Steel Series headset that I take with me travelling, since the rest of it would be a lot to drag along and it has served me quite well without any need of software either. I run Garuda Linux (an arch derivative) with KDE, if that matters - I would hope that wouldn’t make a difference but included it for complete info (not just because “I use Arch, BTW (sort of)”.

¹ - I have not used these personally, but they were recommended to me several years ago by someone whose judgement I trust. I hope they have not gotten shitting since then.

There really isn’t any difference between in Bazzite or SteamOS in my understanding.

This is not true. Not saying one is better than the other here, but they are quite different really - for a start one is a Fedora derivative and the other is an Arch derivative. They are both aiming to fill the same purpose (Handhelds and HTPC), but how they get there is significantly different imo.