If you actually need to get to DXB, then I would just make my own arrangements. If you have travel insurance, the flight cancellation protection might be triggered. But QR aren't going to do much for you, and they are certainly not going to put you on BA - that option ceased to be available a few months ago now.

In order to trigger EU261, the flight cancellation needs to have been within 14 days of departure. If they have already removed CPH-DOH-DWC from their system, there's nothing left for you to be claiming EU261 over. Whether the underlying ticket remains live or not is irrelevant if there is no space held on a PNR.

You said "Amex have confirmed with me that the booking has not currently been cancelled." I doubt the QR segments are still there, so do they just mean that HEL-CPH segment is still alive? In which case, see above.