Swapping plates on a trailer isnt that bad if its what gets you home. The plate is still registered in the OP's name.
This is absolutely a matter of STATE law. For example, here in Connecticut, misuse of plates (swapping) is much worse than unregistered and will probably yield you a court date and a hefty tow bill, rather than a written warning for something minor like towing an unregistered trailer (trailers here do not need to be titled or insured).
The original post was about pickup at the base. This is also specific to each facility. There is a whole section on the Steel Soldier forum regarding this and pickup at specific facilities.
I've only done it once (Mechanicsburg, PA) and it was a breeze.........if you are prepared. All the information on the specific facility is provided through links on GL. Have all your documentation in order, if you have a passenger, they must also have all the same documentation (and may also need to be declared when you make your appointment). Expect to have your vehicle searched...........ie. make sure it is completely clean. Mine was not searched, but other individuals just attract attention.

Arrive early, at least an hour before your appointment, to get access to the base and arrive at DRMO on time. You have an appointment for a reason, you'll be on a list at base security. You and the vehicle will be issued a pass. Just because you have an appointment does not mean you will be loaded at that time. I had to visit several offices/locations on the base before arriving at the loadout location. Do not be in a rush or expect an in/out experience and you won't be disappointed. I waited roughly an hour after checking in with the GL personnel near appointment time, they were cool in that they let us roam the yard to check the future sale items, and find our unit. From arrival to departure I think it was around 2 1/2 hours.
Sure loadout personnel are helpful, they picked up the trailer with a lift, set it exactly where I wanted it on the car trailer, collected a signature and moved on to the next rig. I would not expect more.
As far as towing it out.............if you would do the same buying a car or trailer from a complete stranger who can tell you
nothing about it's operating condition, sure, go for it.

Personally, for me a car trailer was a smart choice.