Sounds likean air leak in the system
Assuming that you have no leaks downstream (pressure side) of the pump, I say I've seen this behavior before.
In my case the pump is installed after the galley foot pump. When the pump is running, it is sucking a bit of air through the sink spigot/foot pump. The pressure pump sucks the air/water through and compresses the air, some of it going into solution. As the air goes into solution, the pump starts up again or just keeps running.
Given that you describe the system spewing air/water, the most likely cause is the sink spigot. If you don't have one, look for a crack/split hose between the tank and the pump where air is being drawn into the system.
Note that if there is a blockage in upstream (intake) side of the system, you will hear the pump load down and strain to draw water.
If it is the sink spigot/foot pump, you can disconnect the foot pump but that defeats the water conservation benefits. Alternatives are to add a shutoff valve before the spigot or as I have done, put a plug/cap on the end of the spigot.
Good luck.
Paul