Wouldn't it work better to pull the hot air from the upper levels and expel cool air in the lower level to force air flow from bottom up?
I have a somewhat similar problem in my single story house, all the returns and vents are in the floor. It's great in the winter, but in the summer the floor will get 50° while the thermostat still struggles to get the rest of the house down to a comfortable temp.
__________________
|