I have already posted Willens’ account which in no uncertain terms explains how the SBT originated. It was not due to Tague. It was to explain where the bullet went after it exited JFK’s throat. [Edit: this theory began in February 1964]
It's possible, even likely, that the FBI knew they had a problem prior to April 1964 but the public becoming aware of Tague's injury is what forced them to come up with an explanation for the throat wound.
The public hadn't seen the autopsy photos or Zapruder film at that point so there would've been little doubt about the FBI's findings had the James Tague issue not become part of the public's awareness...