Congrats on the new trailer and welcome to the forums! Bad news is you are over your ratings on the Santa Fe. Look at page 5-53 in your
owners manual, max trailer weight for the V6 is 3500 lbs but max hitch weight is 350 lbs. The Zinger you bought weighs in (according to the
brochure) at 3286 dry weight with a hitch weight of 524 lbs. Since it's a smaller trailer you will likely only add 500-600 lbs of camping stuff, but then you're looking at a loaded weight around 3800-3900 lbs. Once you add a battery and fill the propane on the tongue the hitch weight is likely around 625. This is without any water on board.
One guy posted here a few months ago that he bent the rear trailing arms on his Santa Fe towing a popup. I think the issue with the bouncing you are experiencing is not so much the total weight but the weight on the hitch.
If you want to get more precise info, you need to get the loaded trailer to a scale. First take the Santa Fe by itself and get a baseline. Then go back with the loaded trailer and weigh the SUV and trailer together, then the trailer by itself (still hitch to the SUV). Subtract the weight of the trailer by itself from the total combined, then subtract the weight of the just the SUV and your answer will be the amount of weight on the SUV hitch.
Once you have facts and data you can decide what to do about the SUV....but there's not much if anything you can do right now to fix an overloaded condition.
One thing you can do is read the
sticky at the top of this forum on how to set up a hitch, many dealers don't do it correctly.
One last point, don't attempt to lighten the hitch weigh by moving cargo to the rear, if the hitch is too light the trailer will sway and the whole caboodle will wind up in the ditch.
