I'm pretty certain what you'll have to pay extra will be only for airport taxes. It may cost more if that carrier normally charge YQ/YR in addition. For my BKK-HKG return in Y, I got charged for just under THB 3,000. (In this case, the currency is THB as the itinerary is ex-BKK.) It's for CX-imposed YR and airport taxes. No booking fee from QR's side (and I hope it would stay this way). Not sure if there'd be any difference when it comes to J/F, but I think extra costs shown on ba.com would be a good reference, if not exactly, for what you're going to pay.
I'm not sure how they calculate required miles for routes with stopover though, because my oneway BCN-MAD-DBV on IB was calculated not by combining BCN-MAD + MAD-DBV segments as I initially thought. The miles required were less than that. So you might save some miles with some creative routing, I guess. I only have to pay extra for a small amount of taxes on this route.
The new partner redemption engine on QRPC's website can now let you select specific flights (they just list the flight, doesn't reflect actual availability), so I'd say you'll have a pretty good chance getting a successful booking if you make your selection according to what's available on ba.com. However, I heard CX's availability for partner airlines redemption goes black after their Asia Miles enhancement. Not sure if it's still the case now.