If you have a 4WD your max is 4,430 and the 2WD is 4610. Looking at the 3.55 gears the limit is 7,420. So that gives you an idea of why you have a low rating. It probably isn’t the truck frame, the tongue weight (likely around 10% of the 7420 number) or payload, it’s the motor and specifically gears.
What does this mean? It means you’re taking a pretty big gamble with a trailer that size. The 3800 pound number is “dry”. This means that is the weight of the trailer with nothing in it, no water etc. Your camper will weigh a fair amount more once loaded up. It’s not just what you put in it. You have battery weight, full propane tanks, etc. I can almost guarantee you’ll be at max and that will be with all water tanks drained. Once you get into the hills and wind, your transmission is unlikely to last long. It will get hot and it’s unlikely you have any heavy duty oil cooler/transmission cooler etc. What I’m getting at is you will need a different tow vehicle to get a travel trailer.
I’ve towed a lot of things over the years, but a travel trailer is the one I’ve always felt I needed to be most careful with. I have a 34 footer that is 7,000 dry, but around 8,000 loaded. It’s a big sail and that is a really big factor that is often underestimated by people. It’s a completely different animal than pulling equipment or vehicles. Without massive wind resistance you can play a little closer to the edge of the envelope, but it’s almost universally a bad idea with a travel trailer.
What are your options and what do I recommend?
No one wants to hear they need a different vehicle, but you’re pretty much limited to a pop up style camper with your setup. So that’s one option to think about. I’ve camped in pop up’s and they work pretty well.
The good news about a new tow vehicle is if you want to stick with the 3,800 pounder, you have a plethora of well priced options. Any full size pickup other than one with the smallest motor and lowest gear ratio will work. Most of the mid size trucks like your rangers, Canyons, or Tacoma will get the job done if it has the factory tow package. Any full size SUV will be plenty adequate if it has a tow package. So there are many options available.
Weight distribution hitch/sway bars and a vehicle rated for the weight. That’s what I’d recommend. The risk simply isn’t worth it otherwise, so a pop up would be the other option. Good luck and enjoy! A camper is probably the best buy we’ve ever made!