I was thinking the same things as Kyle and Deb. The French door is generally a fair weather type of item, in my opinion, and not very practical; they are best kept for summer patio entrances and going between kitchens and solariums, et cetera.
You are way better off getting a regular sized door, and framing in a small wall to fill the gap.
Since it's south, you might want to get an insulated exterior door with a window (double paned) in it, and put a window in your wall.
Not a huge
project, but one that will require some thinking and doing for sure. If you are unsure about doing that sort of work, it's a relatively small and quick project for an experienced builder to do, if contracted out- like a few hours, after assessment and getting the materials.
Apart from that, getting better seals in place might help a bit. Though French doors are much harder to get a good long lasting seal every time you close it unless they are very high quality.
I would definitely go with a regular sized house door, especially in a tiny house that you want to keep warm. The ratio of space to loose heat from in a tiny house, in relation to the gaps that could exist in a French door, make the French door a poor match.