I have the Mac mini m1, the MX master 3 and now also the MX keys plus keyboard. Everything works great via bluethooth, so far absolutely no problems have occurred. I can only recommend the new Logitech products!

I have also installed the Logitech control software 'Logitech options', where you can make settings for the mouse and keyboard. MacOs Big sur up to date ? other devices in the room that could interfere ?

Try connecting to a Logitech USB receiver (if you have one, MX Master 3 Mac doesn't come with one!).

What doesn't work for me via Bluethooth is the Bose Soundlink mini, this speaker loses the connection shortly after being connected to the Mac. I have now connected the speakers with a cable, then it works.

Unfortunately, I also have exactly the same experience (old and M1 MacBook Pro with me). Unfortunately, switching the MX Master 3 off and on again doesn't help. Killing the Bluetooth service on the Mac doesn't help either :(
-> I am currently restarting the MacBook and would also be grateful for a better solution.