Try to completely remove BA from the mix and press them to put you on AA metal from LHR to PHX any way possible, noting the added connection they gave you and 6 hours added to your original trip. Push it up to a supervisor. If that all still fails, then invoke the OW Liaison to get BA space opened up for you to go to PHX in a more desirable routing. Maybe even via LGW so you don't have to switch airports. Something like LGW-MCO-PHX or LGW-LAS-PHX.