Speedbird,
Your fundamental error is assuming the QNH at airfields A & B changes. In the real world of course it's going to but for your theoretical question to work you have to assume they remain the same. Look at Lightning Mate's diagram and just picture those columns of hot and cold air as columns of changing pressure. The rate of change is higher in the cold, dense air but the pressure brought to bear by those columns of air is the same at sea level because the hot air, while less dense, has a taller column (with the same altimeter 'pressure' reading) for all the molecules to whizz around in and exert pressure at the Earth's surface.