I think you can book JL F online with Avios; certainly the availability is visible online. It's fabulous but no 241 can be applied, of course.

You could try LHR-SIN-HKG, use the 241 on the LHR-SIN portion (you can do this as long as it's a BA issued ticket, but I'm sure experts will be along to advise).

If you do want to LHR-PEK then that works fine, and you can transit without a visa to a flight on a separate booking without problems (and stay for 72 hours); just have a printout of the e-ticket for the Beijing Transit desk. Beijing is worth at least a couple of days to look around and the comparison with HKG is interesting in many ways.