I can see why you would want to do this. It might work, but I have concerns. Usually this sort of thing isn't done for safety reasons.
How do you propose to connect it? Higher end
wood stoves usually have a connection for outside air. This is to provide external air for combustion so the stove doesn't pull in cold air through every crack in the house, potentially even affecting other combustion appliances with a negative draft.
Even with a tall chimney and a high end stove, the same negative draft can occasionally happen with certain wind conditions, with the stove struggling to maintain draft. This could potentially push combustion gases down into the ductwork. Not good.
If an insurance adjuster notices a physical connection between the
wood stove and the ductwork, it's an easy excuse to deny a claim.
Personally I would add an efficient bathroom fan at the cold end of the house and push air through the cold air return to a vent beside the stove. I think that would be more effective.