Thank you for posting this question. I have a 2015 Silverado 2500HD and had a similar experience when I purchased our 5th wheel. We have been towing a 3 horse gooseneck with no issues, then got the 5th wheel and wow, what an awful tow. I honestly think it is the short bed of the 2500 that causes the chucking especially with the larger trailer. Your's is much longer than mine so probably much worst. I looked at the Anderson hitch for the light weight and ease of taking it in and out of the truck but want to try one before buying. My concern is the lack of positions it can be moved to. I believe you can move it up and down but not forward and back. I might be wrong, but that is what it looked like to me.
I have the B&W companion and at first I put it as far back as possible and tried to level the trailer with the positioning of the hitch. That was awful for the chucking. During a trip to Yellowstone, every night I moved the hitch to find a better position for the best ride. I had to move it forward quite a bit and my trailer is not perfectly level but not bad. I also found that if my fresh water tank is full, it rides better. Probably pulling some weight off the hitch. It is definitely a balancing act as it is not perfect, but I would guess it is the trade off of having a short bed and short wheelbase truck.
Good Luck, I'm sure you can find the balance without buying a lot of hardware to mask the chucking.