i pull a Roo 23SS. its weight w/options is almost the same.
i have a '07 Avalanche with a 5.3L v-8, w/ a 7200lbs tow rating.
i don't have a auxiliary trans cooler but i think i might get one.
the Avy has the factory tow package with a tranny cooler. but towing in the heat and hills has raised the temp up in the 220 degree range, once or twice. but only for a short time.
i would recommend getting the additional tranny cooler. you're near your max, so i would consider leaving the FW tank empty. that would save a lot of weight. fill up when you get to the campground.
for the little difference in weight, i'd still get the 233S
by the way, welcome to the "Roo Krew"!!!!