With no ability to manually download and install system updates as user-installed packages in Big Sur, getting the bug patched might be tricky. The bug might not manifest when the system requests a different version number, so perhaps an update to 11.2 will bypass the buggy requests and therefore be able to fix the bug
in 11.2 without intervention. It would also be possible—if extremely hacky—for Apple to sideload the update into an unrelated patch that doesn't affect the OS version number.
All else failing, a complete reinstall of Big Sur from the App Store should get you to 11.1—although it's a 12.6GB
download, and as with any major OS installation, you should be absolutely certain you've got good backups of all your data prior to the attempt.