UPDATE: I went back out and was trying to engineer a way to bypass this check valve and noticed something interesting. There is three places to connect a hose to this camper. 1. The city water connection. 2. The fresh water tank fill. 3. The black tank flush.
What I thought was interesting was that the only one of the three connections that didn't have a check valve was the connection for the black tank flush. It was also the only connection that had a screen in it. So I thought hmm, what if they just put them in the wrong spot when they installed them at the factory? So I removed the two connections and swapped them. I can now add water to the fresh tank with a funnel. I went ahead and put some water in the fresh tank to make sure it wouldn't affect water pressure, and everything worked fine. So for now, until I can get a system to add water with an external pump, I think this will work.