Sorry, but that's just not how git works. You can't exclude older commits, and doing so would change the commit hashes and thus break future git fetch operations in some fashion.
I'd recommend improving your backup process if at all possible. As was mentioned before, you ideally only need to back up your Moodle dataroot, and database. You may also want to backup configuration files too if you don't keep these in any form of configuration management system.
Andrew