On Gm pickups, the best thing to do for batteries is to buy dual post batteries, and if the side post bolts are getting rounded over, pull them, and the rubber boots, and put some stainless 3/8 bolts in their place. get long enough bolts you can thread a nut on all the way and then hand tighten the bolt till it bottoms out in the battery, and then tighten the connection with the nut.
Aka "head studs" for your battery.
As far as the GVWR, the truck is 9000 IIRC, and the trailer will be 21000 with the tri axle, so you are over 26K, and all the legal eagles will suggest you get a DOT number and medical card to haul a tractor to a show.
My suggestion is blow the port and play dumb, and if worst comes to worst and you get pulled over, be apologetic, let the DOT officer have his power trip, and try to get out of there without being ticketed. Then sell the trailer and buy one with 2 7 or 8K axles.
Tow ratings are based on axle and tire ratings, but also braking and accelerating.
Key word there, accelerating, so while you may be well within axle and total chassis ratings, you may be over the tow rating, since it takes an extra mile to get up to 80 mph.
My pickup of choice is the 90s GMs with the 6.5L diesels and 5 speed manuals, and of course with 195 hp they dont accelerate terribly strong, so their tow ratings are a couple tons lower than a 454, even though the chassis, axles, brakes, and tires are the same, so for that reason I dont take tow ratings very seriously.
From a safety perspective, I heed the tire ratings, axle ratings, and stay within the designed GVWR of the truck and trailer so that it can handle the weight and stop it properly.
I usually run around 16-22,000 combination with my 6.5Ls, (tow rating is 14,500) and they handle it great.
On your 02, what is the powertrain, if its an auto are you using the tow/haul mode? (if its non-working, its likely a broken wire in the shifter, either splice wire or buy new shift lever) and does the engine have the check engine light on? GM puts alot of powertrain protections in place, and the computer is not afraid to cut fuel if it sees something wrong from the sensors.
I have a buddy that bought a 8.1L powered 01 and it had no power, and wouldnt even bust the tires loose in first. all he did was change the fuel filter, and change all the bad sensors it had, and now it will roast tires through 3rd on the 5 speed Allison automatic.
Also, if it has the allison, go to an Allison dealer and buy a few little red spin-on filters for the transmission. that filter is a full flow element, so its crucial for the transmissions filtration. the same filter is under 10 bucks at Allison, and 30+ for the same filter from GM.