ART - Unless you count the couple millon miles I drove semi's, I don't really have that much towing experience with light trucks, mostly my F250 4X4. Since I ordered it out special I got the added rear over-load (helper) springs from the factory and the camper/towing pkg, truck was already wired for the brake controller. I've towed to many PD's, took SON to/from school down in Lafyette, IN, hauled DAD's '51 FARMALL M home from ILL, towed cars on tandem trailers, etc.
I have the Prodigy brake controller, comparable to your P3, even grossing over 17,000# with the M in tow I had excellent brakes on a rented 12,500# cap trailer. I pulled a 6' X 12' enclosed U-Haul trailer a week ago and I HATE them, heavy built trailers, but they all have surge brakes, and they pull harder than HECK. I averaged 13.8 MPG over a 400+ mile trip down & back to get the M, and only averaged about 12.1 MPG towing the U-haul with the 982 in it a week ago. Only other difference was the route I took, to get the M I ran I-39 down to I-80 then west, fairly flat ground. Week ago I ran Rt 151 to Dubuque then Rt 61 south to Davenport, many more hills, but also 7000# less weight!
When I first loaded the M & loader years ago I drove the bucket all the way up to the front bulkhead and chained it down. I towed about 4-5 miles to my Brother-in-law's farm. I was WAY too heavy on the hitch of the trailer. SON & I moved the tractor back about FOUR Inches, chained it down again. Next morning we went to town on our way home and weighed the truck & trailer, "axled everything out", weighed frt axle of truck, both axles of truck, both axles of truck with trailer attached, then the whole truck/trailer & load. Even though I was about 1400# over what the truck was rated to carry I was not over-loaded on any axles, tires, the truck, the hitch, or the trailer capacity. It towed great, I had about 10-12% of the trailer weight on the hitch, exactly what is recommended.
I towed a bit with my SWB reg cab F150 4X4, it also had ALL the H-D springs, shocks, sway bars, rear over-loads, etc since I ordered it out too. I pulled a 12 or 14 ft tandem axle trailer with it only about 6 miles to move into our house back around 1991. Some of the loads were up in the 4000-5000# range, both my #72 & 129 CC's, bunch of shop tools, etc, plus the load with all our appliances, fridge, freezer, washer, drier, etc. Even though the truck was only rated for 5000# trailer weight, had the 300-6 & creeper low 4-spd it handled the weight fine. Not sure how it would have cruised @ 65 MPH on the interstate but on our blacktop county roads it did fine. The 300-6 has a great durability reputation but even the EFI motors were NO power-house!
I'd start with Kraig's folk's Weight Distribution hitch, and air bags. For most places you'll be towing your brake controller should be fine as long as you have decent brakes on your trailer.
DON't skimp on any of the components like the receiver, drawbar, etc. I installed a Putnam XLD 15,000# receiver on my truck before I towed the M. I bought a solid forged steel drawbar w/2-5/16" ball to pull that trailer, plus the Prodigy.
With the smaller truck, with lighter rated axles & tires you'll have to be more careful on weight distribution, the adjustibility of the W/D hitch will help with that, let you transfer weight off the rear truck axle to the frt axle. You'll have plenty of HP to pull the weight, might not hurt to add an extra trans cooler if your Suburban doesn't have a H-D one.
Couple of the trailers I've pulled, even the tandem axle trailer I hauled our '88 Mustang on one morning I didn't have working trailer brakes, up to around 3500-4000# gross trailer weight I don't really get that "puckering sensation" when I step on the brake and get no sense of slowing with my F250, and IMO, the brakes on my F250 are nothing special.
On another forum I lurk on sometimes I read a post yesterday about "1/2-T towable RV trailers", they had 1500+# tongue weights and weighed 10,000# trailer weight. You only want to tow half that weight. With proper equipment it should be easy.