I can see rules against putting a pop up screen room on the grass, but am curious why they would not be allowed other than that? I was asked to move mine at Silver River SP because I had it on an area under trees that was neither grass or the hard packed surface of my site - think of what the ground looks like in the woods. I happily complied, but have never run into this rule otherwise. My screen room does not have a canvas bottom and guy lines used to prevent it from blowing away.
I do what the OP suggested he was planning - drag the picnic table into it, or drag it over the picnic table.