If your stopover has to be Dubai you should probably fly Emirates.
However, the only Chinese destination they service is Hong Kong. And
if you are prepared to fly into/out of Hong Kong then you have a vast
choice of airlines and stopovers: you might consider Gulf Air (with
spicy food) since you could stop over in Muscat, almost certainly the
most interesting destination in the Gulf region.

There are few obvious places to stop over on the route to Beijing.
Moscow is fairly near the half-way point, but that would mean flying
Aeroflot ... A stopover elsewhere in Europe would make for a long
flight to/from Beijing (even longer to Shanghai) but you could consider
Vienna using Austrian Airlines or Budapest using MALEV -- on this trip
the flight from Budapest to China is actually operated by Hainan
airlines, a reputable Chinese carrier. You can book the MALEV option
at www.zuji.com

Lots of other stopover possibilities with indirect flights -- eg fly
Ethiopian Airlines to Beijing with a break in Addis Ababa!

You could always use some of your BA miles to pay for an upgrade on
their non-stop LHR-Beijing service -- that is the way that I would play
it.