Log burners give off a large amount of heat in a relatively short time. If you have doors open in unheated rooms with cold walls and the significant heat from a log burner suddenly comes into contact with those walls, you get condensation. In our previous bungalow we had a very efficient log burner that I'd had installed in order to heat the entire property. Trouble was that once the heat from the log burner had built up in the lounge and I opened the door into the unheated bedroom, the walls began streaming with condensation. I cured the problem by having a radiator installed in that bedroom so that the temperature was maintained at a level where condensation couldn't form when the door was opened to allow the heat from the log burner into the room. I had a simultaneous problem with condensation in the loft because of excessive heat from the log burner rising through the ceilings and loft hatch and coming into contact with the cold underside of the roof sarking, but I cured that by increasing ventilation with soffit vents as well as cutting down on the use of the log burner.
If you plan on continuing to use the log burner, then you really do need to find a way to heat the entire property to eliminate the conditions in unheated rooms that create mould, and that probably means using the CH.