There is no reason not to book it as two tickets using two different mileage schemes. If you wait until arrival in Doha to book the second leg though, you run the risk of it not being available. The issue with it being double miles will be down to married segments. It's a pain, but I've seen this same thing on LHR-DOH-DXB before too. Presumably availability to DWC is the same?

The only problem with separate tickets would be checked luggage. They may or may not through check if tix are through different ffp's. But if you are HBO, you can avoid going landside by checking in online before departing London (another reason to book the second leg in advance).