Well if you are talking about the exhaust fan in the ceiling of the bathroom.....no there is no relationship other than both devices use 12vdc to operate. The picture in your original post is a furnace heat vent for the bathroom I believe. So that one should blow air when the furnace blower fan is running and heat when the furnace fires up.
Strange that the limit switch is now showing normal zero ohms (could be an intermittent problem with it. If both your sail and limit switches are normal then when you turn on the furnace power and set the thermostat to call for heat; propane on; your furnace should fire up OK. It is probably a good idea to have an RV service tech check this one out for you. Some mobile techs are very good if your Dealer is booked solid. When you find your problem please let the forum know as this helps others learn too.
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