That trailer can certainly be pulled by most 1500 series trucks but it's getting on the upper end. For a more enjoyable and easier towing experience I would look at 2500 series trucks. I realize that will exclude your favorite brand.
If you are going to be putting on 20,000 miles towing every year diesel might make sense. Otherwise, for occasional towing I think a gas engine would serve you well.
In my experience a Chevy or GMC would probably ride and handle more like your Toyota but that's all just opinion and personal preference. You'll need to drive them. That's the fun part anyway.