"I informed him that he could either show us where our assigned site was or I would find it myself and park with or without his permission. "
While I understand your frustration you were indeed fortunate that his next call was not to the local authorities. I'm sure they would have been happy to find a place for you to spend the night. :R
What would the authorities have done? No one broke a law here.
My calendar shows July 4th to be on a Tuesday this year. The 5th, 6th and 7th are Wednesday, Thursday and Friday.
We left Yellowstone on July 2nd last year and arrived home in NJ on the 11th. We stayed in private campgrounds all but one night (which we spent in a Forest Service campground) and had no problems getting a site each night or issues with having to stay or pay for more than one day.
July 4th is a Thursday in 2013 and the OP has a valid concern as that Fri-Sun will be considered a holiday weekend.
They may have better luck with a reservation as we get 30 days or closer.