The GA has final authority for who can board, document checks, and the assignment of seats, including issues such as who is permitted to sit in exit row seats, so I would think that getting past the check in agent wouldn't necessarily be definitive.

IIRC BA is one of few airlines that comps a seat for an infant who turns two during the trip, although this policy might have changed.

In the OP's case, I'm not sure whether a lap infant must be (strictly) under two or whether flying on a lap is still allowed on the second birthday. Does the rule say under two or over two?