The TM backup will suffice, in part. Unfortunately, going back from BigSur to anything else is hard to do. BigSur not only sets the Macintosh HD to read only, it also encrypts it so that it cannot be messed with. As in Catalina, there is then a Macintosh HD - Data Volume where your data is stored. That is the same in BigSur.
So, to go back to Catalina requires that you completely erase the internal drive, wiping out everything and going back to a bare naked drive, then repartitioning and reinstalling Catalina on it, then using Migration Assistant at the first boot (after the Welcome setup and before you create a user account) to migrate your account from the TM backup to the internal drive. It's a real PITA, but it works.
Someone may come along and suggest a clone backup. That approach won't resolve the need to repartition/erase the internal drive, but it will give you a working system as you do that. You could clone the Catalina system to an external drive, put it aside, then install Big Sur, give it a go and if it works for you, fine. If you don't like it and want to go back, boot from the Catalina clone, repartition and erase the internal drive, reinstall Catalina and then clone back from the external to the internal. About the same steps, but with a working system as you do it.
Don't let that difficulty bar you from pressing on, however, Big Sur is, for me, much "snappier" than Catalina was on my MBPr mid-2015 system.