So you flew BA IAD-LHR and QR onward to SLL via DOH.

Surely the issue here was with BA?

It's fairly well known that BA no longer interline bags, even onto other OneWorld carriers, so I'm unsure who told you that you wouldn't need to collect your bags at LHR and take them over to T4.

Also, FWIW, travel class bears no relevance with how your bags are delivered to you once they have disconnected.

M