You will probably exceed your rear axle weight rating (RAWR). Let's do the approximate math.
RAWR, based on new trucks on website. 3,950
Approximate weight of rear empty 2,100 - this is probably close but estimated
Max left to reach RAWR. 1,850
Pin weight per brochure 1,300
Battery and propane 100 usually not in brochure weight
Weight of 5th wheel hitch and rails 150
5th wheel storage is in front of trailer axles
and about 60-70% ends up on your
rear wheels - 1,000 X60% 600
Total est weight on rear axle 2,150
Looks like you would be about 300 lbs over your RAWR (and probably tire ratings if P tires) before you put any people or cargo in the truck. Some of my numbers are estimated but I think pretty close. You could take your truck to the CAT scales and see if the empty rear weight is less than my estimate.
That said, if you buy LT tires with enough capacity and add Timbrens or air bags you will probably be OK on shorter trips. It will take a toll on your vehicle. I wouldn't spend the money when it's not the real answer.
If you can afford it, it's time for a new 2500/250 truck of your choice. I would not pull your new 5th wheel with the current truck. It's just not safe IMHO. Good luck.
