Whether it's required by TSA policy/procedure or not, I wouldn't use a TSA lock to secure the gun case itself. Last time I flew with a gun the suitcase had a TSA lock but the gun case inside was secured with a non-TSA lock. I also used a cable lock to secure the gun case to the suitcase's frame.

FYI, this was a week or two after the FL airport shooting. I had to pick up my suitcase at the lost luggage office, but it wasn't zip-tied or otherwise secured beyond the locks I put on it. I flew Delta.