As a veteran of gaming on Linux for several years, I have to admit I keep a small collection of various usb bluetooth dongles, because honestly, built-in bluetooth support still remains questionable and unreliable in many cases, at least for me and the systems I use it on. I don't necessarily blame Linux as much as I blame the manufacturers of the chips and devices, but unfortunately we have to live with the chaos that their reverse-engineered-firmware-reliant devices create. Any cheapass bluetooth dongle is probably fine, the cheaper and more ubiquitous it is, the more likely it uses the same shitty chinese chip that all the others use and that a bunch of someones already hammered out drivers for, but honestly even with multiple different models and brands it still seems like a crapshoot which one feels like working properly at any given time, but usually one or the other will work and get things to connect, and it's usually perfectly reliable once all the drivers have loaded and it's all paired up and things start working. The struggle is real, though.

Not me, sorry. On desktop Linux, I'm always wired, and the bluetooth always just worked when I needed it on Bazzite or Steam Deck, connecting via the controller setup in the Steam menu, but maybe someone else here will know.

For what it's worth, this wired alternative is almost identical to an xbox one controller except for the rumble motor, which is markedly lower quality. If that doesn't bother you, it's also less than half the price, and works out of the box in all distros I've tried.

Idk if it’s the solution to your problem but I install xpadneo and never have issues.