I tested the idea with a small blower fan. Well I say small but it's a decent sized and can move a lot of air.

I put it at the base of the stairs and by itself it caused the air temperatures to normalize between the bottom floor and the middle floor.

But it's also super power hungry and I don't want to burn 300 watts of power an hour 24/7 to normalize the Air temps.

There is a central wall that goes from the top to the bottom in the middle of the dome and I believe there is a pocket that I can utilize to hide the ducting.

I guess when I finally execute on the plan I'll take some pictures and show off my handiwork.