Airpods Pro disappear from Bluetooth after switching users on Big Sur

On Big Sur, my Airpods Pro disappear from the Bluetooth devices and stop working when I switch user accounts. They reappear when I logout and login again. This is obviously really problematic because I have to close everything and logout just to get them working again. I have seen other people ask the same question with no answer. Has there been any resolution to this?

What I want to happen is that when I switch back to a user account that was using the Airpods Pro, they reconnect and become the active audio device again. A second-best option (one that would mean MacOS was at least "not broken") would be that the Airpods Pro still show up in the audio devices list and Bluetooth when the hardware is available.