Do nothing. Get to the airport about 20 minutes earlier than you otherwise would, but say nothing, just hand over the relevant paperwork. Probably you would be fine anyway, but at worse the airport can have a go amending the ticket once its under their control. At this stage, it almost certainly falls into the "too difficult to fix" category.

The formal responsibility is with the issuing ticket airline (BA? AA?) but the ability of the other airline to fix something is very difficult. In the lead PNR is started on AA, it creates the shadow PNR on BA, but it is well known that the AA to BA to AA interface screws up names. Doubling up middle names is a favourite outcome.

Your mistake, incidentally, was to use a middle name at all. When travelling you should stick to your main surname, as spelt in the passport, and your usual first name. Two names, that's all. The big problem is patronymic and matronymic family names from the Spanish world, so airlines and airports are well versed in this area.