Well, first the special TP for RV's is not the best to use, Scotts single ply or Angle Soft rank rather well. And second, I seriously doubt that's the source of your clog.
If it is in fact a paper clog filling the black and gray tanks both, then opening first the gray, then once the flow is established, closing the gray and opening the black AT THE SAME TIME is the best way, this puts max pressure above and the flowing water, wishing to continue flowing, will SUCK on the outlet. I do not suggest this as a first attempt though
However as I said, I don't think that's your problem.
Another possible trick is the Flush King. (At least that is what I think it's called). This is a special fitting with a hose fitting so you can put water in and a dump valve to keep from simply going down the hose.. Hook up with this in the line, and a clear adapter above it to the tank, and of course the hose to the hole in the ground. Leave all dump valves open and by adjusting the knob on the flush king direct a jet of water up into the tank.. If it's TP this will chew through it fairly fast. If, it's what I fear it is, neither of the above methods will work. I'll let others fill you in on the fear (There is a thread on it here somewhere)