I've been mulling over why Captain Brown may have come up with the story about the X-1 tail being modified. I wonder if he has conflated his story with this event.
On the 5th June, 1947 an open house was held for about 100 members of the Aviation Writers Association at Muroc. Airframe #2 was used for a ground run demonstration of the engine, which resulted in fire damage to the tail due to an explosion in the oxygen side of the number three cylinder head. The aircraft was returned to the Bell factory at Buffalo for repair on June 9th, where #2 was repaired by the simple expedient of installing the tail taken from airframe #3, then under construction.
Airframe #2 was returned to Muroc on the 27th July.