My new PS5 controller connects just fine via USB, however, after a few minutes of being connected through Bluetooth, Steam says that the controller has been disconnected and the controller stops functioning in-game or on the desktop. The white light on the controller stays active (which I think means it is still on), and according to the Bluetooth section in System Settings, the controller is still connected. If I disconnect and reconnect, it works again for a short while, and if I plug it in via USB, it works perfectly.
Not sure if I provided all the necessary information, please let me know if anything more is needed!