Probably not much help to you, but we had this dilemma last year when we changed car, the dealer was keen on us having the 1.6 T-GDi (174bhp) manual box . I had no doubts it would have the power, but my main concern was the kerb weight, at 1604kg it was lighter than I would like. I was sure it would tow my Elddis two berth, but I wasn't convinced it would suit something bigger should we change. We decided to stick with the diesel for the extra weight and low speed torque and in our case it was the right choice.
Not a Kia but I towed a covered car transporter (so about the same physical size as small caravan) that weighed about 1100kg all up. The cars I used were a Golf GTi 112 bhp (kerb weight 970kg max tow weight 1200 kg) and a Golf TDi 90 bhp (kerb weight 1100 kg max tow weight 1300 kg). The GTi never had a problem, totally stable and the engine had enough power. being 2WD it was a bit challenging on grassy paddocks but you learned to park in certain areas. The TDi certainly benefited from the additional torque, the loss of 20 bhp made no difference. Again totally stable and still a bugger on grassy paddocks.
Should also add that I carried a fair bit of weight in the car, tools, spares etc.
I can see no reason for the T-GDi to have an issue, however you wallet will suffer with the mpg compared to a diesel.
Extra low 1st gear makes towing easy. Yes, it changes gear 'more than without the extra load' but that's what a 7 speed DCT auto does.
Coachman Laser twin axle 'van or 5.8m Humber Destroyer 120hp/6m Tornado 130hp RIBs.
'Manual' option on overrun uses engine braking downhill without issue.
Towed similar for several years with Land Rovers, Santa Fés, and other vehicles occasionally as well.