I strongly recommend that you do your research on paper, and then rent 2-3 different types of RVs and confirm what you think from your research.
Different SUVs have different towing limits. A Platinum trim level Ford Explorer has a 3.5L Ecoboost engine and 5000 pound tow rating. Ford Expeditions is 5900-6200 lb or higher with a max tow package. Tahoes and Suburbans are in the 6000-6600 lb range depending on the exact version. Toyota Sequoias are at 7000 lb.
Generally you want to be around 80-85% of your tow rating, and properly set up with hitch and sway control, for a good towing experience.
You might be able to find a good deal on a used class B or C, and the cost would not be much more total than a tow vehicle and trailer.