I booked a ticket for myself and a friend of mine through a 3rd party, with one leg in the US, the second leg is an international segment. My friend's name, who is on the same itinerary, is Jane Smith-Mitchell (pseudonym) according to her passport. However, when the ticket was booked, they put Smith as the middle name. The booking confirmation as it came through in my email looks ok (Jane Smith Mitchell) but when I select seats it says Mitchell, Jane Smith.

I caught this later on but the 3rd party will not correct the mistake, insisting it will be fine. Should I be worried?