They changed the firmware so that it can "seamlessly" move between devices but 1) it's not seamless at all - I actually disabled the new behavior and 2) I'm pretty sure something else changed regarding Bluetooth because switching between devices manually is slower and less reliable than it used to be before, and I bet this change is also responsible for the issue with input devices that you're seeing.
After upgrading to Catalina ~10 months ago the MS mouse has been flawless in daily usage. I haven't gone back and tried my Logitech mice.
https://www.reddit.com/r/apple/comments/anvufc/psa_2018_macs...
It's basically not completely reproducible and is random.
Having said that I've got a USB key which if I use it everything on Bluetooth stops working.