You have several tight constraints in your specs...
The tow vehicle may be insufficient for a family of four towing the trailer size you are looking for. The 6000 lb tow capacity is one thing, but you will also need to consider payload, which SUV's are usually a little light on capacity. Everything in the SUV and trailer tongue weight counts against payload, so wild guess numbers of 120 (x4) + 720 tongue = 1200 against payload assuming you attempt to tow a 6000 lb loaded trailer.
Re: trailer weight and dealers - dry weight will allow a rough estimate, but loaded for travel weight is much more important. Shipping weight from manufacturer, or dry weight after batteries, LPG tanks etc can be significantly lower than an average loaded for travel weight. I have seen many add in the 1000-1500 lb range.
DO NOT rely on the dealer to determine what your vehicle can tow.
Lastly, a 4k price range sounds difficult for what you are looking for, but I haven't shopped for a used rig in a while. Well used trailers may require some repairs that can get expensive or time consuming if you DIY.
Were I in your shoes I would come up with a trailer weight you can reasonable expect to tow based upon the TV specs and go from there.