I have configured my system to be something like you outline above.
My thoughts were to merely run the hot air blower from the furnace for air circulation. Although not circulate "all the time", but when the stove room reached a predetermined temp. I reversed (rewired) an old thermostat, which I had, but an "air conditioning T-stat" would be what was needed. I tied this to the "fan" circuit. I have hot air ducts, as well as returns in the room. However, they were "add-ons", so I don't know how well they work. (The hot air was there, but I added some returns, through the floor joist/ceiling of the room.)
So, if the room gets warm enough, the T-stat turns the furnace blower on. The problem, which I think you eluded to above, is that the moving air feels chilly, as it's fairly low temp.
On my furnace, I believe the blower has to be wired for "fast or slow". For this application, "slow" would be preferred. .. for heat, "fast" would probably be the choice.
I'm sure this can be overcome with some relays, etc., however to complicate it, something should be incorporated to switch the blower back to high, if furnace heat is called for during the "cooling cycle" of the stove room.
Addendum: While the above method sounds like it should work and be a method, which would incorporate existing devices, I'm not sure it's the best.
One reason, the room has to get pretty warm to be able to realize the heat gain at the other end of the house. Maybe slower air movement would help. ??
Right now, it seems like a box fan, set on low, is doing a better job, to keep the room temperature down, while sending the heat off to the rest of the house.