gate keeper error when trying to install Monterey update

Boot into Safe Mode according to How to use safe mode on your Mac and try installing from there.

NOTE: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

What version of Big Sur exactly ?

Attempting to upgrade to Monterey from say a Big Sur 11.6, 11.61. 11.62, 11.63, 11.6.4 etc can be problematic.

IMHO - upgrading from the Latest Version 11.6.8 Big Sur should not create and issues as per description in your question.