This is another case where I feel we have to really to get down to details before deciding as this section of plumbing has so many different small points that vary from one RV to the next!
A leaking check valve on some might let the pressure down and make the pump cycle, but one thing to watch on that point. If it is letting pressure drop, it is likely due to water going out of the line at some point, so when we are saying a leaking check valve at the fill port, it seems likely that would show as water being forced out of that port. We should spot that as a leak, right?
But if we have a case where line pressure is being reduced enough to cycle the pump and we don't have a leak, there is another way we can be having frequent pump cycles, depending on which form of valve we have at the fill port.
Click these snips to get a better view or go direct to the parts and see if this makes sense to you?
The first snip is the overall area where the line comes from the right as I've marked in red and down to the line passing the valve.
Then for closer detail, I zoomed in and continued the red line marking to pass the valve and then the line goes back to the fresh water tank!
I think this is one where the problem is not the check valve but the valve which we change to fill the tank or use as normal in the RV.
But if we look carefully at THIS RV and forget what other RV have, I think we can spot the problem.
My fav for help on parts is here:
https://catalog3d.winnebagoind.com/menu/Parts.htm
If we drill down to the OP RV and go to plumbing and the supply side, we can get a good look at the tank/pump and lines.
Looking at the drawings and following the lines, I propose that the cause is NOT the check valve on this one as they report no leaks coming out there.
But this RV uses just a simple cutoff valve rather than a diverter valve and IF that valve is not sealing well to cut off water, it can come out of the tank, to the pump, and on to the valve, which is leaking and go back to the tank again in one big continuous loop without ever showing up as a leak to the outside where we might spot it!