Given the sheer volume of international to international connections across Oneworld LHR I suspect you may me misguided.
The majority of which is incredibly low yielding. I flew from OSL-LHR-HKG in Club World for £110 more than it would've been to take the same LHR-HKG flights originating in London in economy. BA do not rely on connections in the same way that KLM and Lufthansa do for the majority of their routes, hence why they were against a third runway - on top of what you said, they know they don't need that extra 'hub' capacity to remain profitable.
Also you say "double capacity at LGW" to aid p2p when there is considerable free capacity at LTN/STN/SEN for inbound leisure. You don't NEED to build a new runway anywhere quite yet for that market, regardless of what GIP might want.
Ignoring SEN, those two
are reaching capacity, particularly for based aircraft. There is very little capacity left for peak time based slots/parking positions in the whole of the South East, hence why every time Ryanair or easyJet open a new base overseas, the first route to change is usually the reversing of the morning London route.
Besides, a new runway isn't just for now, it's for the future - as we know we're not getting another one any time soon