Hi, I'm not sure if you found a solution. I starting having terrible lagging issues, along with shaky movement, with my Mighty Mouse 2. It was working perfectly one day, I charged it, and then it suddenly became an infuriating experience. I was very close to taking it to the Genius bar.
I did all the things you find out there, including:
Cleaning it
Removing bluetooth preferences files
SMC reset
Factory reset
Resetting bluetooth module on Mac
etc. etc. etc.
None worked.
I suspected Wifi interference, but I don't have many options in terms of moving my router and computer. I thought that maybe there was bluetooth signal interference, maybe?
I'm only typing this here because I found the solution (and others can try it if they are searching for help):
I have two external monitors for my Macbook Air. The way it handled the dual monitors after the update to Big Sur changed, so the mouse issue may be more related to the OS.
I opened the Display preferences and set all my monitors to "Scaled". This fixed the issue immediately.
System Preferences > Displays > (Change from "Default for display" to "Scaled")
Does that I mean this is a graphics problem/lag? I have no idea. My mouse now tracks perfectly, you have no idea how happy I am now. This problem was creating constant stress and I am so relieved it is solved.