Yes you will need to get it registered. Harbor Freight will give you the paperwork you’ll need. Here in Texas you get it registered at the tax assessor-collector office.
I’ve used the harbor freight trailer for a few months now, which include a ~1000 mile trip. No complaints at all. The rear track will hang off a little but as long as you have it strapped down tight, you shouldn’t have any problems. You could also build a wider platform if you don’t like it hang off the sides. I’m planning on doing that and maybe a second platform for another kart.
I did not enjoy putting it together (hot a$$ garage) and probably would have paid a little more for something pre-built. One issue I had was a poor ground for the brake lights. Even though I had continuity, one of the brake lights would not work. I had to sand off the paint at the back of the trailer where the brake light bracket meets the frame.