My suggestion is to create a new APFS Container (created while booted to Catalina) on your internal drive outside of the Catalina and Big Sur Containers. Copy (by drag and drop) your home folder to the Volume in the new APFS Container.
Open System Preferences→Users & Groups, unlock it by pressing on the padlock, select <your user name>, control-click, select Advanced Options..., for Home directory: click on the Choose... button and browse to the location where you copied your home folder. You will need to reset PRAM after you make the change, this part is important so don’t skip it. Now boot into the other OS and update Home directory: and reset PRAM.
I would suggest not removing the user account home folder in the Catalina and Big Sur partitions just in case you encounter some issues with the shared home folder.