Could a 'damaged' TimeMachine backup still be of any use?
The nature of Time Machine is such that it isn't an archival backup, as files deleted from the source drive will eventually disappear from the Time Machine backup set.1 If you have no other backup however, it might come in handy.

If the sparsebundle can be repaired, you'd be able to extract files from it. Following the instructions in the first green box of http://pondini.org/TM/A5b.html might let you access the backup set.

If you do gain access to it, you can use the Time Machine interface2 to remove all backups of folders you don't care about, but it might be tricky to cause the sparsebundle to take up less disk space.3


1 See the text starting with "Time Machine keeps" in https://support.apple.com/HT201250
2 "Remove a backed-up item from your backup disk" - https://support.apple.com/kb/PH11241
3 "Changing the size {…} of a sparse bundle" - http://pondini.org/TM/A8.html