Avoid updating fields in the MoveTo procedures. The main reason is that it will cause a lock to be created in the database, and the lock will not be released until you move focus to another object. When someone else puts focus on such an object they will get an error message. Another reason is that if you for example copy, move or rename an existing object you may end up with inconsistencies.
If you have access to Site Administration, you could run a SQL update for all existing test sets:
UPDATE TESTCYCL SET TC_USER_02 = (SELECT CY_CYCLE FROM CYCLE WHERE CY_CYCLE_ID = TC_CYCLE_ID)
To handle renames you might run this SQL at regular intervals, or you could add a button in Test Lab that uses the OTA API to take care of all cases where there is a mismatch.