If I were in your place I would definitely use the Shogun for the towing. There is a lot more to it than maximum permitted towing weight.
A major factor is the rear overhang of the towing vehicle: The tow bar should be as close as possible to the rear axle to ensure stability, and this is a feature that you tend to find with off road vehicles.
I towed caravans for over 30 years, covering most of Europe, and often visiting the Alps. Only once did I get into a "snake", where the caravan starts to sway uncontrollably. That was with a Jeep Cherokee, which had quite a long overhang and was lighter than my usual Range Rovers. I immediately replaced the Cherokee with a Nissan Patrol, which towed so well that I often forgot that I was towing!