If I were you I would try this: Boot into Safe Mode. The goal is to free up as much disk space as possible so hopefully, with minimal load, you will have enough free space. Now you must download macOS Big Sur without triggering an automatic update which could use up much of the free space you now have.
Instead, download the installer manually. Still in Safe Mode, launch Terminal and enter the following command:
softwareupdate --list-full-installers
I’m assuming you still have internet connection. A list of macOS versions should come up, all that are compatible with your Mac. (I can’t bring up Big Sur on my Studio because in will not run Big Sur.)
If the list loads and Big Sur is on the list, now enter the following command:
sudo softwareupdate --fetch-full-installer --full-installer-version 11.7.1
and hit return and enter your password, assuming Big Sur 11.7.1 is on the list. It should now start downloading the installer into your Applications folder. 11.7.1 should be a shade under 13 Gigs, so check your available disk space first. When it’s done, it will not install automatically. First make a copy of the installer (“Install macOS Big Sur.app”) to an external drive, for future use.
Then you can either reboot, still into Safe Mode, or straightaway launch the installer from the Applications folder without bothering to reboot. The important thing is to perform the installation in Safe Mode, and if successful, address the other issues from, at least, a fresh installation of macOS. Good luck.