Water pump cycles on and off

[sourdough]

If your pump is running 2-3 seconds then only stays off 3-5 seconds you have a fair sized leak you should be able to find. You said you've purged all the air. If it was the toilet that much water could be heard and seen exiting the toilet. If it was elsewhere by not that much water should be evident in the floor or running under the trailer. That assumption is if it was running all the time. If you are shutting it off after each use then the water may not be accumulating that much.

I would first check every connection; under every sink, toilet, drains, water heater, outside shower; behind the walls...everywhere. I've found leaks over the years in/under all of them. If there is nothing there then pull the pump and see what's what.