Your problem is not unique. I'm guessing that you will have noticed how the changes in temperature are synchronised with the behaviour of the water pump, warm when the pump runs and hot when the pump stops. This is more common with caravan's with inboard diagram pumps but it has also happed to some high flow submersible pumps also .
This happens becasue inside teh water heater is an air cap - its designed to be there and its job is to give cold water space to expand when its heats up. The down side to this is the pressure in the air cap also raised becasue its compressed by the expanding water, and its hotter.
The pressure is released when a hot tap is opened, as the water is expelled teh pressure drops and eventually the cold water pump will kick in a few seconds after the tap is opened.
If the pump can deliver more water than is being used, it will begin to replenish the hot tank until the pressure switch is satisfied, That stops the cold flow, but the air cap in the hot tank will again deliver hot water, so you can get an alternating hot/warm cycle.
This is usually worse with showers becasue the mixer tap effectively connects hot and cold together and the rose provide more back pressure which causes the hot pressure to feed back to the pressure switch via the mixer tap exacerbating the problem.
There is a solution and that is to fit a surge damper to the cold water pipe work. This mirrors the action of the air cap in the hot tank. Whale sell such a device, and Truma sell a combined damper and pressure switch. There may be other manufactures - your caravan dealer should be able to supply the parts .