I would not want to do it. You truck could probably crawl down street with that trailer, but you would need a WDH and Brake Controller no matter what.
A modest Weight Distribution Hitch and brake controller alone are going to cost you ~$300-$400 and you do need these to tow the TT. Then the WDH would need to be installed/setup with your TV and TT present, the brake controller could be done ahead of time. By the way the "Tow Package" are things like the receiver, transmission cooler, 7 way plug, etc...the WDH is altogether different. The WDH replaces the standard hitch ball and ball mount that you insert in the receiver for towing light weight boats/trailers.
If it was me I would call a tow truck service and see what they would charge to move the trailer. If it is only a few miles I would imaging they would do it for $20-$30 each way. Also, since you are technically a customer of the private campground and use their storage facility too, they too may tow it to the the State Park campground for you on occasion, for a fee of course. There is a similar set up on the Central Coast in CA where an expensive private beach front campground is next to a State Park. The private CG storage facility is down the road further from the SP CG, I have heard if you pay to use their storage they will deliver your TT to the State Park for $30 and another $30 to retrieve it.
Just a thought.
2007 Chevy Duramax LMM/Allison
2012 Jay Flight 29QBH
2012 Jay Flight 26BH (Sold)