If a normal RV furnace is letting combustion gasses into the RV interior, the heat exchanger is shot and the furnace is unsafe to use and needs repair or replacement. Indeed, all other things being equal, using the furnace would decrease the relative humidity inside since the ratio of actual water vapor (unchanged) to the maximum noncondensing amount in the air (which goes up with temperature) is decreasing. The only RV propane appliance that would increase inside humidity (and sometimes quite noticeably) is the range. If you do have an unvented propane heater, that also would increase humidity inside.
The humidity monitor might be spiking because the sensor doesn't work very well at lower temperatures, or because something else is increasing the relative humidity (something damp that's drying out as the moisture that was formerly in or on it evaporates), or maybe even because the humidity sensor itself is cooler than the air that's being heated and so has some condensation forming on it. In the last case, the reading would correct itself over time.