I think a bit of caution should be used on the Best Price guarantee. My suspicion is that Travelup are selling consolidator and special fares, not public fares. So they will have their own conditions and rules which means a direct comparison could be deemed not to be met. For example the BA.com fare may allow changes for a fee, the private fare will not.

That said I really don't recommend these OTA bookings unless you are a thousand percent sure of taking the flight. And if there are AA legs on the trip then even then I'm not convinced, due to their frequent alterations.