It is talking about electric brakes on the trailer's axle and you installing a digital proportional brake controller in your car to control those brakes in proportion to the dynamic inputs you give the tow vehicle, specifically the degree to which you're braking. Since your car was not INTENDED for such an installation, you'll have to run all the control wires yourself, both under the dash and to a new 7-pin harness at the back of the car. Not impossible, but not easy for a novice. By comparison, I had a plug-and-play harness already under my dash and a 7-pin connector already installed next to my hitch. Easier installation, to be sure.
I think you'd have problems locating a 1,500# trailer that has brakes installed since they are not required in any state for that low of weight. Generally speaking, you need to head north of 3,000# or 3,500# before brakes are required.
You could add them, of course, at your expense and trouble, but as others have identified, you to NOT have the optimal tow vehicle. Trash it? Don't know. Not good for it, I can be sure of that. The transmission just isn't contructed with "towing" being a major part of its duty cycle. Stop/go traffic is where you'll really heat up the fluid. That is assuming you achieve lockup of your torque converter under 'normal' circumstances. If THAT doesn't happen, I give you about 400 miles of towing before detonation. I'm kidding a little in that...but only a little. You'll cook the fluid in no time flat.
As far as what to buy? Any 1/2 ton pickup or SUV would easily tow the widget you're describing with aplomb. I'd start by going that route, setting your budget, and seeing what's out there. The newer, the better, generally.
Even if you can find something I doubt you’ll be happy with it, your Hyundai just wasn’t designed to be a TV. You seem inclined toward small RV’s so if you could keep the car and pick up a good used ½ ton PU as a TV, even with the smaller V8, It would easily pull the smaller stuff and you’d be much happier. And it would open up your options a lot as to what you can tow.
Finding a new RV with low VOC may be difficult. Finding one in a small RV may be impossible.
I find they all have "new" smell. I am quite sensitive to that and noticed it myself. That new smell is what will bother someone with allergies.
So, a couple years old may be better to already of "aired" out. When I bought my brand new trailer, I put my Oreck air purifier in it for a couple weeks. Really reduced the new smell. After that, just open the windows when its dry out and let nature air it for you. Especially before a trip, or when its overly hot out.
As the others say, find a small RV you like, then pick out a vehicle to tow with. You may also want to consider moving to a Class B camper van or small Class C motorhome. If you have to buy another vehicle to tow with, and want to keep the car for daily driving then this may be a good alternative. The motorhome or van will have less insurance costs than a pickup or SUV. The car will certainly be cheaper as a daily driver... you could even tow the car along with you so you don't need to tear down camp to go site seeing once at your destination.