@TECK, does booting from 11.3 to Mojave and back work consistently for you?
Let me try that, I have not tested. In the past, I would lose the Bluetooth when I would switch back from Mojave to Big Sur, remediated by an additional reboot. My current setup requires Esc key pressed to display the bootpicker. Will report back in few minutes.
Edit: I can reboot back and forward between Mojave and Big Sur, without issues, I tried twice only. Every time, I experienced the previous behaviour where I would lose Bluetooth in Big Sur, forcing me to reboot again in order to gain it back.
I attached a quick video which shows the progress bar login in Big Sur, with Bluetooth not functional.