United looks like the best bet for you.

My advice would be to fly a route like this:

DFW-EWR-FCO on UA
Stopover
FCO-ZRH on LX
Take trains from Zurich to London
LHR-DUB-ORD-DFW on EI and UA

Flying LHR-DUB-US instead of LHR-US will let you avoid the high APD fees at Heathrow.