I will start by saying I am not familiar with either your caravan or the details of the water system, but becasue the shower housing (pic 1) has no electrical contacts, I will have to assume that your pump is controlled by a pressure switch which will be located close to your present water pump.
This would explain why there are no wires close the the shower out let.
I'm not certain that you can simply swap the shower outlet for a pump input, becasue they are doing exactly opposite things, and the pipework may not be connected to allow it to receive water being pumped into the system at that point. There could be check or Non Return Valves that may prevent reversing the flow of water in the showers pipework.
If however your idea is correct, then you would have to put in a pair of wires in parallel to your current pumps connections to the new water inlet terminals.