Often1 this is a QR matter and BA has nothing to do with it.
OP has purchased a ticket IAH-DOH-MLE. That is what he has reservations for. Due to flight cancelations IAH-DOH QR purchased a, probably deeply discounted, BA ticket as part of rerouting to regain the original journey. Note QR 'own' that segment not OP!
OP has tried to claim with BA Quote " I've tried the normal, "missing Avios" route but it won't accept the ticket number as, again, it's QR stock, " with no luck because he is trying to claim points on a ticket 'owned' by someone else.
He has a contract with QR for transport IAH-MLE and he will receive points according to that contract. Be those points Qpoints, Avios or even raindrops it is QR who award them.