Sealing the clean out door does not mean the rest of the space below the T is sealed or air tight. If the old liner tiles are missing, cracked or broken, chances are the masonry isn't tight either. Unless the T is sealed tightly around also, anything open, cracked or untight above it will also let it draw cool air in also. Sorry, just a bad idea, again, as it has already proven to be issues with some folks on here that have had installers, "leave the T cap off" and have issues directly related to it.. Doing a half assed job anywhere on a wood or coal burning system is not something that would happen in this home, or suggested to any one else's.
Feel free to rig your set up any way you want, but giving poor advice to others, that may potentially cause them issues or worse, is not a good practice. With these appliances, it is best to practice overkill, than cut corners.