Let's see what we have here. City water connection works. We can assume plumbing is in pretty good shape. Have no idea of the year/model of the camper since it is not in the signature. Don't know if you bought it new or used.
Does this trailer have a water pump winterizing kit on it? This would be a valve on the suction side of the pump to draw antifreeze in to the pump instead of water from the supply tank? Sounds to me like a supply problem for the pump. The check valve in question here, won't keep you from pumping water normally, it is there just to maintain pressure when there isn't any water flowing. The pump should pump water without a check valve, and I think most are inside the pump. Find the pump, and look at both the inlet and outlet side side hoses. Maybe there is a valve there.
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