BA F on the 747 (I assume that’ll be the plane to CPT) can be a bit underwhelming. In terms of space I find it a notch better than on a QR 787 (DUB-DOH), but IFE, screen, table etc. is better on QR. Depending on your travel date you will probably either have a 787 or a 777 on the DOH-CPT sector. While the 777 product is not bad, in particular if travelling with a companion, I would still prefer the 747 BA F.
I find service to be consistently good in QR J, but hit and miss in BA F.

Ultimately I would choose BA because of the nonstop LHR-CPT sector. It is more convenient. In my opinion it is not worth trading a day in or around CPT for a day in DOH.

I have found DUB staff to be helpful when checking through, even on separate tickets (which i understand to be against the published policy).
CX F lounge in T3 is a nicer place than the Al-Mourjan in DOH for a layover.