I think that it may be possible, but would be dealt with on a case-by-case basis as outlined above, and at a shorter timeframe than someone who likes to plan (like me) may like. However, it is highly unlikely to be against any written policy, so you certainly could ask the appropriate BA people at the airport. They are a business after all, so if you offer money for a product, the certainly won't turn you away custom if it benefits them, and there is no harm in asking (I mean, some people try to do this kind of thing for free, so there is no problem doing it legitimately in the manner you suggest).

It just may be too late to be a deciding factor for the ticket you choose to buy.