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Geeze, you have a gas fireplace that transfers kinetic energy to the outdoors? Wow!
Now, the damper is the "door" that closes your chimney off to the outside...Are you maybe talking about the chimney? Because I don't see how you pull air "from" a "damper."
You don't want to be pulling air into your house from above a burning gas fire...That goes without saying though so perhaps I am misunderstanding your question. Can you draw a better diagram?
Combustion does give off CO2 so you may pull that in...but you don't want to breathe what's supposed to be going out your chimney.
And I know I'm nitpicking but what is "frostbitten" air? Are you talking about cold air from outside? Why don't you just passively allow air to intake as you EXHAUST hot grow room air directly outside. If you grow room is inside of a heated house, you can just suck air from the house through the room and then out to the exterior...