No, it was the dark side Star Alliance that screwed BA and AA. I know, I saw their plans, but nobody in BA management wanted to talk to me about it

You should look at the different options for WTP fares, as it might be cheaper to nest the tickets, I have not looked. You might want DFW-LGW-AMS return and AMS-LGW-DFW return operating inside it.

Out and backs to AMS from LGW are very easy, with on-line check-in, print your own boarding pass, no immigration at AMS and security done at the gate. As said before, that will get you your 4 BA sectors very quickly.