I plumbed my RV like this:
City water comes in to a T in the pipe leaving the pump. Water cannot flow backwards through the pump, so no water enters the water tank. City water pressurizes your entire post-pump water system and everything works fine. There should be a T in the pipe between the pump and the camper's faucets and water appliances. The T has a shut-off valve. The valve opens the pressurized city water pipe to be able to fill the water tank. From your description this is the only way to fill the tank. When the tank is full you turn off the valve. As long as city water is connected the pump will never turn on because it always senses enough water pressure "downstream", so the RV uses only city water pressure. When u disconnect city water (a have pump on) then pump will operate normally.
#1 you have to have a cut-off valve.
#2 there has to be a vent.
#3 the vent has to exit the camper. It will not dump water into the enclosed under belly of the camper. It will exit outside of the camper. Just look for a hose or an opening where water can exit. There has to be water pouring out somewhere when it overflows. Possible it could dump into your gray tank or into the drain (leads to gray tank), but if so this would be a horrible design and needs to be corrected so the water drains to the ground. There is really no other possibility I know of. The water sould drain to the outside, plain n simple...