I am glad you resolved it to your satisfaction
RoarAhead.
KVS - I don't doubt your knowledge of the BA fare system, but for the benefits of future readers there is an English Law doctrine called Estoppal.
There are several specific types of equitable estoppel. Promissory estoppel is a contract law doctrine. It occurs when a party reasonably relies on the promise of another party, and because of the reliance is injured or damaged.
BA could be argued to be estopped from denying that a passenger could purchase an infant ticket for 10% of the cost of their ticket. It is for this reason that it would need to be escalated - to someobne who could approve an infant ticket at the 10% price regardless of any established Fare Rules (as quoted by you). If this could not be done because someone sufficiently senior could not be found, then one would have to pay the inflated cost and claim for the difference through the County Courts. I would be confident of success in those circumstances.