Remaining cylinder pressure at BDC

Is the reason there is less cylinder pressure at BDC on a higher compression ratio engine, purely down to the reduced temp? After all, at BDC, assuming the same cylinder filling efficiency between both engines, you would have the same volume of gas in the same space.

You're correct. The higher compression ratio also results in a higher expansion ratio which allows more energy to be released from the expanding combustion gasses.

You also have to be careful to compare like to like - the pressure in the exhaust port will have at least as much affect, as that is resisting the evacuation of the exhaust gas. For tuning NA engines this can make a big difference, as can manifold pressure for turbo' engines.