That pump does not have a simple check valve, it has a bypass valve.
The manual describes it -
The by-pass is a spring loaded diaphragm that opens up allowing water from the discharge side back to the inlet side. The by-pass is set to begin opening at about 40 psi and creating full by-pass at about 62 psi
The manual describes how to adjust it.
So, the valve may be working exactly as intended, but the city water pressure is high enough to open the bypass valve, filling the tank. Do you use a pressure regulator on the city water line? What pressure is it set at? It could also be leaking.
It's interesting that more people don't report similar problems - that's a common pump and common RV pressure regulators provide 40-50 PSI. So if the bypass begins opening at 40 PSI as stated, one would expect to see this happening a lot. I wonder how common it is for manufacturers to put check valves between the pump and tank. As mentioned earlier, that should fix the issue.