The operational case for LHR expansion is widely accepted. However, the financial cost of providing LHR R3 is extraordinarily high, at close to 100x the inflation-adjusted cost of delivering the second runway at Manchester (£273M). Many, myself included, argue that this extraordinary sum does not come close to offering value for money. The choice you desire cannot be provided without due consideration to cost. The (very large) public element of funding required would be far better spent directly on long-neglected infrastructure projects in the regions, for example dealing with the bottleneck that is Lime Street Station.
This debate is comprehensively discussed on the PPRuNe Heathrow thread. Go back about three months and read to date if interested in this particular topic.