You control where the air comes in
Sure, weatherizing the doghouse will minimize the amt. of cold air infiltration at the Fp IF you can seal it well. One of the best sheathings to retain the insulation and provide an air barrier is Thermoply with the seams taped with foil tape. Don't forget under the Fp in the cantilever.
Mikey, you need to provide some air in that house--period. If you don't like where the house chooses to bring it in, it will decide for you. If the leaks around the doghouse are the last major leaks and you seal them, you could backdraft a water heater, furnace, boiler, or open Fp.
Open a basement window until the cold air quits coming in around the Fp. That's how much air you need to bring in. You may need to install a powered makeup air system. I agree w/ JTP on the pilot along with the rest of his advice.