Great to read. Inverting the first name/surname is an extremely common mistake with internet booking and is usually handled routinely and satisfactorily by all airlines. Problems arise when the fare bucket you used is exhausted before you can make the adjustment (rebook and cancel).
AF customer service (at booking stage) can be impressive. But it can also be less satisfactory. In general I find the call center for elites quite good and efficient. And if I get less satisfactory service , I just call again to get another agent. The call center for internet booking is also quite good, but sometimes involves "complex" technical operations on their side; but they try hard.
My experience with BA has always been good too.