If you're going to use bmi miles you might consider coming all the way home with them. Assuming you can book the return to zone 2, it's only 18,750 + £127 +tax to come come back in J, something like NRT-HKG-LHR//LGW-BGO (and you might miss the last leg). And you might be able to book the shorter NRT-HKG leg in F. As an alternative to buying the miles a credit card each will get you 20k miles, but if this takes a while to come through it may limit availability.

Getting out there, assuming you're into your new Amex year there are ways to hasten the arrival of your 241 - book a refundable ticket - and despite the written rules you can book a oneway LHR-NRT online. Not the best use of the241 but if you'll be too short of miles to use it anyway this might work for you. You might also have enough miles left to get something out of your July 241: a quick trip in CE?