Had your same situation and needed the DB clean last week. There is a stored proc in the DTA db named something along the lines of “purge all completed transactions”
I just run that as needed then shrink the log. Its a good way to clean the DB while still running in production when you dont have an option on taking your servers into maintenence.
Its relativly lightweight and although youll see it drop performance a bit, youll still be moving data at a good speed.