This is an alternative answer after trying to do the same as you are,
I did install a floor register with a small built in fan but it moved very little air upstairs. Very dissapointing.
So I installed a small 24 inch low profile ceiling fan at the top of the stairs in the hallway so I could pull the heat up the stairwell
and distribute it down the hallway and it worked fantastic, on warmer days I could shut it off and colder days turn the speed up.
I thought I would pass this idea on as for about the same amount of work you get 20x the air flow. I was lucky because I had a Light in the hall and
was able to replace it with the ceiling fan.