I just don't understand people who want to run AC in the winter. Y'all like paying power bills or something?
The problems with bringing air that cold into the room are, in no particular order, wild temperature swings, wild humidity swings, cold spots and of course where the exhaust goes.
4kW is too much for ambient cooling. It's almost enough to heat your house with. Your best move is to move the heat, if not the air itself, into your home. A closed loop air to air exchanger would do it.
Next question; how will you get adequate CO2 in there?
Finally, how do you plan to manage humidity and keep the temperature from dropping too much during lights off?