No way to avoid LHR when going to CPT on Oneworld unfortunately.
The best timing with the shortest layover would be via AA 80 leaving DFW at 9:20pm, arriving in LHR at 12:25 the following day and then continue on BA 59 to CPT which departs at 7:30pm. Travel time is a bit less than thirty hours, layover in LHR is seven hours.
AA 80 is on the 772 (which doesn't have MCE) and the BA flight is a 747.
Personally, I'd take the earlier AA 78 to LHR and trade two hours more in the BA First lounges in LHR (you state you're EXP...) for the 77W and the relative comfort of its MCE cabin on the DFW-LHR segment