I went thru something similar regarding the need for a cdl when I purchased a large boat.
I got quite a few different answers even from the folks who are enforcing….. I called the Wisconsin dmv. They told me that under 26000 lbs, don't worry. When I pressed them for written confirmation, they told me to call the State patrol as they would be the one ticketing me. I called the State patrol and they said I should be fine. I told them I need to be 100% clear and legal. They said they would check on it and call back. An hour later, they called and confirmed that it is not what you are towing that matters, it is the combined GVWR of the truck and what is being towed. If that number is over 26000 lbs, you MUST have a cdl. So, my truck is rated at 12500 lbs, boat trailer at 22000. No question, I need a cdl.
I did tell them that this is for personal use, no monitary gain.
By the way, my actual weight loaded is just a bit over 25000lbs. Again, the GVWR trumps the actual weight. So, I asked the nice officer if I could ask a question. Say the wife and I went out for dinner, each had one beer, and get stopped for a trafic violation. The officer smells beer on my breath and I blow under the legal limit. Will I get a ticket for dui? He says no. I then ask, if I tow my boat and I am under the 26000 lbs threshold, will I get ticketed for being over weight, even though I am not. He said yes! The reason being is that I could be. Might be different in the states you are traveling. A friend of mine was ticked for exactly this reason, and he was not loaded. Personally, I was glad I got the cal a few years back, going thru the process here in Wisconsin is now kinda expensive as you must go to classes.