I have the exact same situation. I have Big Sur installed on an external drive, Catalina on internal drive. Big Sur boots fine and my password at login works fin, but I can't unlock padlock in System Preferences.
Dazzzz
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This has been happening to me several times a day since Ventura RC and now with 13.1. It's nearly unworkable. It happens when I Command-Tab switch from a full screen app to another. Often safari to mail or vice versa. I had Stage Manager turned off as I like to work in multiple full screen apps, but I turned Stage Manager on today and reduced all of my apps to no longer be in full screen mode and it hasn't crashed so far.
I also have a fully specced 14" MacBook Pro 2021 with M1 Max, 64GB RAM and 2TB SSD. So it's not lacking in grunt. I have it connected via USB-C to a HP Z27 4K monitor.