Poor insulation = mold!

If you don't heat, the insulation problem will only get worse. Either ventilate or heat (preferably both).

I believe the most pragmatic option would be to run a dehumidifier with a hose connection and integrated hygrometer when absent. So it should only run if the room humidity >50%. Condensate drain hose into a sink or somehow outside, and done. Alternatively, always keep two opposite windows open, secured against break-in with grilles.

A gas heater causes a significant moisture input. This then also soaks into the wood, furniture, and walls. You can't get rid of it by just briefly airing out before you leave.

I don't want to scare you now. But it doesn't look like mold to me. And I don't think it is mold. Mold would not only be at the bottom but also in corners, etc.

This is also not due to insufficient heating but is penetrating moisture. This is a fungus. If you're unlucky, it is dry rot. Then demolition is necessary. The boards get moisture from below, behind, or wherever. You can see that clearly.

If the boards somehow have contact with the ground or if a moisture barrier is missing or defective, then the boards draw the water up properly. This is just beginning. Then lack of ventilation and it starts.

You will have to open the wall and air must get in quickly. Otherwise, the fungus grows and you might have to tear down the whole hut.
The spores of the fungus are in all the wood. And at every spot where no direct air reaches, the fungus grows. It could even be much worse on the backside.

But mostly only between 10 and 25 degrees. So heating while moisture supply from below continues can possibly accelerate this process.
So the colder, the better.

Wait until spring and it feels really comfortable. Very dangerous.

I am also in favor of automatic dehumidification. To answer the question: Insulating from the inside is extremely difficult, especially because the airtight layer has to be executed down to the smallest detail. But then you still have a strong input from burning propane.. Dehumidifiers and switching to a split air conditioner could be the methods of choice.