You’ve said he’s a British citizen, and therefore he is legally required to enter on his UK travel docs. He may have booked his ticket using a US passport but he must provider the UK info for his arrival. In the very say way he must legally enter the US on his US passport.
He can either
1) apply for an emergency travel doc from your local UK embassy,
2) gamble - land, go to British citizens line, present his expired passport along with his US one which _may_ overcome the objection
It would be a bad idea to try to enter on a US passport and not tell Border Force. They will almost certainly know as he is the same person, and they information share on multiple passports. This is there to close loopholes on avoiding the authorities.
