My question is, does adding weight to the back of the trailer take some weight off the truck bed (to keep me in my payload), or does everything add weight to the hitch, no matter where you distribute it to?
Some, yes......but not as much as some would think. Some folks have mistakenly said the trailer is like a teeter totter which it isn't as there is no single pivot point.
Loading out my tandem and triaxle flat deck/stock trailers has shown very little hitch weight change by moving heavy stuff/freight behind the trailers axles. Single axle trailer are different.
What I do notice is adding heavy weight behind a trailers axles makes the trailer tail heavy depending on how heavy the load is....pin to trailers axles.....number of axles...... overhang length aft of the trailers axles/etc.
The tail heavy trailer has a tendency to pull the back of the trailer out on sharp turns or a curvy highway and can be a hand full in strong side winds.
GM gives your truck a 6200 rawr for carrying loads in the bed. Use it.
The 3500 SRW is the same truck as yours except it has a higher rated 18"/20" tires and wheels and higher rated rear spring pack for a 7050 rawr.