That is the way it normally works
First + MI + MR/MS
But even if the TSA is being a bit flexible, if the check-in counter refuses to issue a border pass, you never even reach them. And depending on the itinerary, the initial leg may not even be BA. Unless the airport is too far away, I'd run it by them and see what happens.
Similar case with worst possible outcome, although the mangling was a lot worse:
https://www.tripadvisor.com/ShowTopic-g1-i10702-k9541181-o20-Expedia_and_the_wrong_name-Air_Travel.html
PS. This is one of the reasons why we no longer use indirect bookings for anything anymore. The few pennies saved usually aren't worth the time spent fixing the problems that ensue. (Friends are going through a similar issue right now where JetBlue scrapped one of their 3rd party bookings and everyone involved passed the hot potato for a week. To their credit, JB finally did reissue the tickets and apologized.)
But even if the TSA is being a bit flexible, if the check-in counter refuses to issue a border pass, you never even reach them. And depending on the itinerary, the initial leg may not even be BA. Unless the airport is too far away, I'd run it by them and see what happens.
Similar case with worst possible outcome, although the mangling was a lot worse:
https://www.tripadvisor.com/ShowTopic-g1-i10702-k9541181-o20-Expedia_and_the_wrong_name-Air_Travel.html
PS. This is one of the reasons why we no longer use indirect bookings for anything anymore. The few pennies saved usually aren't worth the time spent fixing the problems that ensue. (Friends are going through a similar issue right now where JetBlue scrapped one of their 3rd party bookings and everyone involved passed the hot potato for a week. To their credit, JB finally did reissue the tickets and apologized.)