I think asking for help was a waste of time. Most of you are quick to criticize but offer no actual help. There are obviously no hot rodders here who have built a car from scratch and had to get a new vin. You also have way more money to spend. I apologize for wasting your time.
The help you don't want to hear you take as criticism. Some have offered ways to improve the situation but there is no getting around the legal aspect of it.
You can add to yur comfort level when towing and increase the safety for towing your combination, but you will never be legal to tow that combination.
Your truck's GCWR is 13,800 lbs (2017 1500 with 5.7L, 8-spd trans, and 3.21 gears). You cannot exceed that total weight rolling down the road. With a ~10,300 lb trailer that leaves 3500 lbs for truck, hitch, driver, passenger(s), and load. I'm pretty sure your truck alone will exceed that weight. Add that the most payload you have (before options) is 1690 lbs. Options, people, stuff, gas, and hitch will eat that up pretty quick before you get to hitch/pin weight - which is part of payload.
Towing capacity is a farce until you get to the HD level. Your truck's legal towing capacity, which is 8470 lbs, is pretty much unobtainable unless you have a stripped down bare-bones Tradesman or cab-and-chassis truck. Swapping to 3.92 gears only gets you to a GVWR of 7235 lbs, GCWR of 15,950 lbs, and max towing capacity of 10,620 lbs, which you won't be able to take advantage of either.
Towing capacity is a capability, not a limit. Your limits are:
GVWR - 6260 lbs
GCWR - 13,800 lbs
You truck cannot weigh more than 6260 lbs total (truck + payload) and the entire setup cannot be more than 13,800 lbs.
You will be hard-pressed to find a shop to certify new legal weight ratings based on your work (or if they do the work themselves). They would be accepting liability and all legal responsibility for any future issues. Even if they did it would not be a cheap endeavor.
Can you get away with your current setup? Probably. Should you? That's between you and your karma. There's more people's safety at stake other than yours. You have asked for recommendations and many have replied. Some have helped you improve your chances of success with your current situation, some have spoken to what they would do, and some have spoken from experience. It may not be what you wanted to hear (especially since your first post said "don't tell me to get another truck") but your request has a lot of legal implications that many have felt compelled to respond to as you will be on the road with us with this setup.
I bought my LD truck new when my old HD truck got totaled (not my fault). If I was going to keep towing anything larger than a utility trailer I would have taken the insurance company's check and bought another HD diesel truck again, no matter how old it would have to be for me to be able to afford it.
Attached is RAM's official spec sheet for your truck.
I'll get off my soapbox now.