Edit: I originally asked this in terms of forcing TM to do a "full" backup. After getting answers about how every backup is a "full" backup, I realized I asked the wrong question. The problem is that I have a backup that I don't trust, on a disk that has other stuff on it, including TM backups from other machines. I'd like to do another backup of one of my machines, independent from the first one. Below is the rest of my original question.
I've looked all over, and there just doesn't seem to be a way to do this. Lots of people taking guesses that seem to work in some cases, but not in mine. I've tried just deleting the folder that contains the backup, but of course the finder won't let you delete it, or even rename it. Bash won't allow it either -- it always says "operation not permitted". Is there really no way just to make TM do a second, independent backup to the same disk without reformatting it?