I personally would not recommend BT for anything other than music and non-gaming content. The delay is too high.
If you have both consoles, there is only one choice to make, and that is the Xbox Maxwell, unless you're ok with using the 3.5mm aux cable.
The Xbox Maxwell (if/while the PS5 accepts general usb audio) will work when set to PC mode on specific usb ports (test all PS5 ports as some won't work due to protocol differences).
Otherwise, both will work through 3.5mm/aux connection to the controllers.