You have to be willing to live with a smaller trailer, depending on the SUV. Look at the manufacturers tow limits, and add 1000 lbs to the empty weight of the trailer. You'll likely discover that you are limited to a pretty small (~17-21 foot) trailer so need to have a very friendly family.
I towed for years with two minivans (Ford Windstar, Toyota Sienna) and a medium sized SUV (Honda Pilot) with no issues, but was very careful to make sure I stayed within the manufacturers weight limits. I used a tongue scale and also weighed regularly at a CAT scale, to make sure I was okay, and loaded very carefully. Minivans have wonderfully long wheelbases and make for a very stable tow platform, although they have low weight limits. The Pilot's wheelbase wasn't quite as good, but had a higher weight limit and stiffer suspension, but had to be careful to not transfer weight to the front axle above what would be there without the trailer. The rear axle in the Pilot is quite beefy, but the front axle doesn't have a lot of margin.
The point of the above isn't to recommend (or disrecommend) those vehicles, but to point out the considerations that need to be taken into account. What you need to consider would depend on the SUV chosen.
Also, plan on a better than average hitch. I used an Equalizer for about 10 years, and then a Hensley Arrow.
I have a family of 5 lightweight people and two dogs. I'm the heaviest of us, at 150 lbs.