Can understand your concern, but with BA, such stories are pretty rare for travel to the USA (Tui has a bit more form with trying to insist on 6 months). Most check in staff will know the correct rule to apply, you'd be pretty unlucky for sometime to try and deny boarding on this basis. If it did happen, all you can do is calmly ask them to check Timatic and escalate to their superiors if necessary, because you know you are correct. Some check-in staff (and even supervisors) have an unfortunate tendency to "double down" when challenged on such things, but you should get there in the end, with some patience.
As posted above, BA would be on the hook for serious compensation due to incorrectly denied boarding, but that wouldn't help you on the day. This is just one data point, but my son travelled to the US with me in April 2023 on BA, with a passport that expired in July 2023, no problem being allowed to travel.
This is the link to the official US government list of countries which do not require 6 month validity on a passport to enter the US, just validity for the intended duration of stay
https://www.cbp.gov/document/bulleti...alidity-update
This list is sometimes referred to as the USA's "6 month club" which is a misnomer since its members are countries that do NOT require 6 months validity, but the correct eligibility is covered in the preamble at the top of the PDF you can access from this link. The UK is on the list, so 6 months validity not required. Perhaps a printed copy of that would be useful as backup at check in.
To my knowledge it is only the EU/Schengen which has the rule about non-member passports (excluding residents) being no more than 10 years old at date of entry. The UK stopped adding extra months to renewed passports in Sept 2018, so this problem is gradually receding, passports issued since then have a maximum 10 year validity. In the case of the OP, though we don't know the age of their daughter, if they have a child passport this would only be valid for 5 years in the UK anyway, so this would not be an issue.