Are you using the Default device selection in OBS? Or in the game that you want to capture? That can and does change randomly, which device it actually looks at, but you seem to have found one of its many triggers, if that is indeed the problem.
Default is good to prove that a fresh installation works, because it's the most likely to be the one that you're using at that exact moment, but for anything else, it's a liability. Always choose a specific device *in OBS*, and in every other program that you want to capture via the Desktop source or Audio Output capture.
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Also, Bluetooth specifically has several problems of its own, in addition to the usual wireless gotchas. Try to avoid that if at all possible, and use a wire instead. Yes, the wire is less convenient in that you can't just jump up and run away, but it's a LOT more reliable!
