Does CarbonCopyCloner work in Recovery Mode in Big Sur? That is, if I buy a 1 TB external HD, format it with a bootable CCC/Big Sur disk image, and then boot my old MBP from this new external HD, should CCC be able to see my files and create a backup? I was able to install Disk Drill via Terminal and it was quickly able to see all of my files, so I assume CCC would be able to, but I‘m not sure. Thanks.
AFAIK CCC needs regular system running, so you cannot run it from Recovery mode. You might be able to mount the disk using Target mode to another mac and run CCC on that mac to clone the disk to external enclosure. But it may still be challenge, my understanding is, that making bootable clone is either not possible or marginally possible in Big Sur. Reports vary and I do not have experience myself. Reports are complicated since now we have intel macs (better chance) and M1 macs (less chance) to deal with, so it is bit of mess...
What I would do is get external drive, mount somehow existing disk - in target mode or in recovery if possible - together with the external disk and copy as much as possible of "Data" drive to it. Ideally all. In Finder or using better tools (if in Target mode) or using command line if in recovery. Once the data are copied and tested that they are accessible, I would try to delete some data from "Data" drive and see, if that would let system boot or at least reinstall. There has to be a lot of junk data somewhere there. Or delete folder with something I have good backup of - music, photos, some large stuff - and see if that liberates enough space. May be delete any snapshots which may be where lots of space is hiding...
If not, I would boot in internet recovery, reformat the drive clean, test if structure is now OK, instal new system and then try to run Migration assistant to move my data back from backup. If it works, great, if not, move data manually and reinstall applications.
Anything above will take time, can nuke the data or whole SSD and in general, is pain to do. Good luck.