It sounds to me that there's a flow restriction somewhere in the hot water line. Here's what I'd do to check. First understand how the water system works. The pump draws water from the tank and pumps it to a tee. One side of the tee is sends cold water to the sink and then to the shower; the other side of the tee sends water to the hot water heater. The output of the hot water heater also goes to the sink and then to the shower. Since there's no pressure tank in the system, the pump normally runs almost constantly when either hot or cold water is being drawn from the sink or shower. Yes, sometimes the pump cycles when you're drawing only a little water but the pump doesn't develop enough pressure in the lines when water's being drawn to stay off more than a few seconds before the pressure in the lines drops and the pump has to turn on again.
First test to see if the cold water pressure and the hot water pressure are equal at the sink (they should be). Turn off the outside shower. In the sink, time how long it takes to fill a container using only cold water; repeat with only hot water. The flow rate should be the same, if not, there's something plugging the slower one (probably the hot, from your description). There are valves in the hot water line between the pump and the hot water heater in some campers that are used to close off the hot water system in cold weather. Check to see if the valves may be defective.
If that test shows pressure at the sink to be the same, repeat the same test with the shower, first the cold and then the hot. If the flow rate's different, then there's a restriction in the line after where the hot water splits off to go to the shower instead of the sink.
It sounds like your shower hot water has pressure for several seconds without the need of the pump. This tells me that you're not getting a normal flow rate of hot water allowing the hot water to retain pressure for longer than normal and longer than the cold water. So then when the pump kicks on, the more freely-flowing cold water overwhelms the more-restricted hot water and chills you.
Let us know what the test results are.