Stelvio 2.0 has plenty of braking capacity, and plenty of horsepower to tow a spider on an aluminum trailer with spares, and is not a lightweight vehicle itself. Focus on trailer brakes, transmission/differential and electrical system, trailer length, tongue weight, total load with spares, weight distribution on the trailer, and avoid lugging the engine. Sounds like you have towed before and know this. I would always want trailer brakes for a trailer over 2,000 lbs. You will technically void your warranty, so don't say you blew your tranny towing a 4,000 lb trailer. Keeping speeds reasonable like towing with anything will ease the strain on an expensive driveline and provide trailer control.
Stealth makes a nice looking hitch, I have thought about it for the smallest airstream trailer. Alfa made a trailer module. Does that include electric trailer brakes? Since the modern cars are sensitive to circuit loads, I wouldn't want to be splicing into existing wiring not only for trailer brakes but for lights. Buy a new AGM battery with the largest CCA, preferably a deep cycle battery. Don't know if the stock alfa tranny cooler has enough capacity, probably does if you don't abuse it and you can probably get a bigger one installed. Not sure about the center and rear diff cooling though, at least change the tranny, center and rear diff fluids.