As noted here you can't access the Time Machine backup from your Big Sur machine on Catalina:
Not only must you use Big Sur to back up to an APFS-formatted Time Machine volume, you can’t even access the backups from a Mac with Catalina or an earlier macOS version installed.
You will have to bring a newer os to your existing machine to temporarily boot it, upgrade (which you expressed a preference to not do) or bring the data to another machine to move it.