I don't really see it as a LHR or LGW decision. Clearly the airports commission was obliged to look at it as such (once the Thames airport was ruled out) but I think wider considerations are at work.
Yes there is significant demand to and from the London and South east. Yes there is also significant demand to and from other areas of the UK.
Why not develop significant capacity elsewhere? It looks like airlines, rather than routing via European Hubs (as had been the argument) are simply putting extra capacity into places like Manchester and Birmingham.
Yes, London is incredible and should be supported. But why not also seek to support other large cities with this type of investment. £16bn is a massive amount that would probably buy you a complete new airport and infrastructure outside of the south east. spending that on heathrow would be poor value for money.
If LHR wish to pay for it themselves - that's another question.