QR
Nice idea chinapattern and I think China Eastern have done something similar by adding HAM to their PEK-FRA rotation, creating a HAM-FRA-PEK through flight. Sadly I don't think this would work for QR going from BHX as the majority of pax will be transiting DOH to go on somewhere else, which makes a (hypothetical example) BHX-WAW stop-DOH-SIN seem quite unattractive in terms of journey time compared to the single connection BHX-FRA/CDG/AMS/DXB-SIN or even BHX by road to LHR-SIN journey.
I personally think Scotland will be QR's next UK port of call (either EDI or GLA) they seem happy for Midlands traffic to route through either LHR or MAN. EK figures from BHX have also been suffering since the big increase in Middle Eastern flights from MAN so is it a case that there is simply not enough demand (at the moment) for a BHX-DOH aswell?