Short answer: Yes. You will want a WDH for this rig. Check your owners manual, and I'll bet it's listed in there for any trailer more than 3500lbs. They probably want trailer brakes for anything more than 1500 lbs (?).
are the specs for current Shamrock (including the 183), which are Roo clones. The dry "ship" weight his listed at 3459lbs, and GVWR is 4806. I'm guessing that the 3600 lb weight that you mentioned is the actual empty weight on the sticker of an actual trailer you have in mind.
Typical recommendation is to plan for 800 lbs over listed dry weight for your actual loaded trailer weight. That puts you at 4259 lbs of loaded trailer; let's call it 4300lbs. From conventional truck math, that would only leave you with 200 lbs for passengers in the Enclave.
But IIRC, the Enclave, like many crossovers and even minivans, actually have a combined weight rating (GCVWR) that allow you to have more in the tow vehicle than just the driver, even when you are at max tow rating. The reasoning behind this is that the limiting factor on a beefy full frame truck is usually the drive-train, because the structure is so stout. But for a crossover, the drivetrain is fantastic if it has extra cooling from a towing package...but the structure is limited. The result is a limited tow rating (what the structure can handle), but an allowance for more in the tow vehicle because of the strong drivetrain.
In the case of the Enclave, IIRC, the GCVWR is 10,400lbs. I remember this becuase I shopped these crossovers before I settled on the Mounty (12,000 lb GCVWR and cheaper). Subtract 4300 lbs, an you are left with 6,100 lbs for the loaded Enclave. I found a base curb weight of 4922 for an AWD Enclave. Not sure what yours is, but lets be conservative and call it 5100. Subracted from 6,100, and you have 1,000 lbs available for driver, passengers, and anything else you put in the Enclave.
You need to check my numbers, especially the 10,400 GCVWR. It's possible that this was the number for the Traverse, which may have had more cooling. But this is the type of math you need to do, when towing with a minivan or crossover. It's not necessarily the same as when towing with a full-frame truck.