OP is an FFL, (guessing newer, since he hasn't dealt with this or anything similar, to apply the logic) he is asking about the process to take care of a customer.
He never said the BATF cared anything about the divorce, nor did anyone else, even though many have said "The BATF doesn't Care"...
From what I can tell, the OP's customer just want the gun that HE bought as a gift for the Ex in his name out, so if the soon to be Ex wife ever sells it or does something stupid with it, and then in the unlikely event that its traced back to him, he can show a paper trail that it left his possession...
No big deal, help the guy out. Personal pref on how he wants to handle it...