I also believe Time Machine's GUI is not able to do that.
But if you feel like fiddling in CLI from terminal, and do some basic scripting, it can be done quite easily using the
tmutil addexclusion and
tmutil removeexclusion would allow to change your exclusion list by script whenever you want.
tmutil startbackup -destination would allow to trigger a backup by script to the desired destination, at the appropriate time (what about just after you changed your exclusion list? ;) )
- script scheduling could be done using
launchctl (or with a basic
Of course, you should disable Time Machine in the Preference Pane, in order to prevent it from starting automatically while your scripted backup may be already running.
Bottom line, have a look at
man tmutil. Backups would still be plain standard TimeMachine backups, just triggered differently. Restore could be done in GUI as usual.
Of course, if you have no previous experience in terminal mode or scripting, this gonna be a little harder. But if you feel like learning something geeky, this is the time to try (and then ask for help if needed!). ;-)