Welcome to the forum.
Did I read that correctly, your hitch is three INCHES behind the axle center? And that is to the center of the pin on the fifth wheel? It does not make sense that 3" could make 500# difference in the front axle weight. It's been a long time since high school physics, but my math comes up you should only have 60# change on the front axle at 3" vs. 185" wheelbase assuming a 3600# pin weight (somebody correct my math if I'm wrong). Even so, 500# should not make it handle as bad as you are experiencing, think about it, 500# is the equivalent of 80 gallons of fuel and I'm sure you don't notice any different handling on empty vs. full fuel tanks. Did you weigh it on the same day with the same fuel/passenger load? What are your rear axle weights with and without the trailer? The difference would be your pin weight of the trailer. Your truck is way bigger than my GMC 6500 Topkick that I tow with, and we are both much bigger than the 3500 dually the average idiot is towing that same fiver with. It should tow your trailer like you literally don't even know it is back there, regardless of where you place the hitch. I'm currently towing a 38' 15k fifth wheel camper, but I spent many miles towing a 40' 22k race car trailer with the same truck. My hitch is an inch forward of axle centerline and my my front axle weight goes up by less than 100# with a 5000# pin weight on the trailer. There are a bunch of guys towing fifth wheel campers with hdt's like yours, and most of those guys have their hitch 3 to 4 FEET behind the axle either to make room on the deck for a car or golf cart, or just to make it easier to back up, without any handling problems. I have to suspect your handling problem lies elsewhere.
Do you know the total weight of your camper and the pin weight? The rule of thumb is 20% of the total weight on the hitch, so if your total trailer weight is 18k, the pin weight should be around 3600#. Water in the camper can make a big difference there, my 100 gallon fresh water tank is well ahead of the axles and makes a big difference in pin weight full vs. empty. There is quite a bit of room for variation there and still handle fine, so anything in that range should handle just fine. My 22k trailer had 5000# pin weight and you literally didn't know it was back there at 75 mph (other than the weight on a grade of course), hills, curves, stopping, no problem at all. Handles like a sport car compared to towing with a dually. And your truck is heavier and has a higher capacity than mine.
Just brainstorming. Is your truck air ride or spring? If is is air ride, does your ride height stay the same loaded/empty, or does the ride angle on the truck (can't think of the term I want) change significantly? If it does change a lot, is there an auto levelling valve that could be bad? Have you gone through the front suspension/steering? King pins, tie rods, steering box play etc. could all give that same symptom which may not be as noticeable empty. Tire pressure? Most tire mfgs. have a load/inflation chart to give you proper inflation for the exact weight on each axle. Too high/low for the weight can make it feel weird, I know my Michelins call for only 75# of pressure on a 110# max tire for my weight, and it does make a big difference in ride. I'm also assuming your tires are in good shape to begin with. What about the angle of the trailer? A triaxle fifth wheel trailer is very sensitive to ride angle. If it is not very close to level, it will definitely handle all squirely, you need to be within an inch or two of level on the hitch height to handle well, and too low will handle worse than too high. Trailer axles bent or out of alignment? Broken springs/hangers on truck or trailer? Just tossing out ideas.