OP - the answer you are looking for can be both simple and complicated.

If your child's passport indicates that he is a British Citizen (the machine-readable passport code shows GBR), then nothing you have mentioned will apply, because the child has the right to abode. In that case, the passport is valid to the very last moment, as soon as the holder is returning to the U.K. At the same time, because the child is a U.S. LPR, the same rule applies as well.

In short - based on the validity rule does not apply when you have some kind of entitlements in the country you are going.

But, if your child's passport indicates otherwise, then you have to research what kind of U.K. Passport and what kind of British nationalities your son has. British nationality law can be quite complicated to those who know nothing. In that case, I will suggest to post back.

True. But you missed an important point - they are not good. Still you have to listen to them, as they can deny boarding based on their limited knowledge in immigration.