2002 Honda Odyssey van, 240 hp, towing capacity of 3500 lbs. 2 children, 2 adults. what can we safely pull?? 2 different salesmen have told us 2800 lbs. is not a problem. What about gear in trailer and tow vehicle? Who do we trust? TT is prefferable to pop up.
Subtract the weight of the children and the adults, then add back in 150 pounds (allowed for driver). Now subtract the weight of all your gear, food, water, tools, etc AND the weight of any accesories added to the van. That is your REAL maximum towing capacity. 2800 pounds will almost certainly be a problem unless you are all real light people, who don't carry much.
Also check that adding the pin weight of the trailer, and the hitch weight, to the van does not cause the van to exceed its Gross Rear Axel Weight.
I don't think a TT will work on that van. Maybe not even a pop-up, unless you can find one with a Gross Weight of 2000 or less and a honest dry weight which allows you to carry what you need (weigh it before purchase, as often added options are not included in the dry weight). Sorry, but except for the smallest of utility or boat trailers, less than 1/2 ton truck equivalency generally should not be used for towing, and your van is 1/4 ton at best.
Even if you can remain within specified limits, it is fairly likely that your engine/gear ration combination will not pull up a hill well.