Any AA flight from DFW to any European destination will have lie flats in business class, and you can build an open jaw itinerary (DFW - XYZ, ABC - DFW). If your requirement is nonstop in both directions, then AA will be your option since BA will only offer nonstop between London and DFW.
And yes, the costs for the YR charges are ridiculous when redeeming points for a BA flight (and other OW airlines), where AA only charges the ~$5 taxes. So, the next issue is finding availability for using your points.
If you can't find business class availability using points on the dates you desire, you may be better off buying the tickets outright with the $3200 rewards cash and paying the extra $$ out of pocket. Or one ticket with points and the other using the $3200 reward $$.
Your time frame for travel (April-May 2022) is far enough out, so be aware if you book something now it will likely have some schedule changes (and even aircraft changes) between now and then.
When building open jaw iteneraries, do you book them at the same time using the "multi city" option? Or book the first leg completely, then go back and book the second leg?