What is the weak link in the carrying capacity? Suspension? Tire capacity? Rear axle capacity? Frame? Something else, a combination, all the above?
Depends on the vehicle but for a ?? model 1500 GM truck the P tires and probably stock wheels need a upgrade. Next would be rear suspension help for more load carrying ability. I've seen specs on a 1500 semi float GM rear axle in the 4800-4900 lb range. Brakes are a function of the trucks axle ratings.
I’m wondering if there is any way to upgrade the weakest component to safely coax a couple of hundred more pounds of carrying capacity.
Sure but again it not generic to all trucks but in a the case of a 1500 truck with a semi float rear axle its pretty limited to my above points.
I would use your trucks RAWR for figuring how much hitch weight the rear axle can carry. Trucks GVWR or the mfg payload or a tire placard number isn't used in any legal aspect to determine any trucks payload. Truck makers aren't about to build a truck frame that can't handle the GAWRs.
Example; Your rear axle may have a 3950 RAWR. Many 1500 truck owners report a rear unladin weight in the 2300-2500 lb range which leaves your truck with around 1450-1650 lb for a max load before going over the trucks RAWR. IMO every truck owner that is concerned with weight should know the truck separate front and rear axle weight numbers.