If you run out of caulk, maybe some thick mud? :/

I was doing some more thinking about this. You mentioned that you can't build the big masonry stove in your house. Can you build one adjacent/under, and then run a big tube of air/water through it and your house? Get some radiant heat going, like you once described to me for a mechanic's garage? I guess that would probably be inefficient as all-get-out, not to mention it would suck if you went to all the effort and there wasn't a noticable benefit or it made the problem worse.