Wow.... my Chevy Express 3500 (15-passenger) is limited to GCVWR of 14,000 lbs. so I'm envious. I have a gaggle of kids (eight) and tow a Keystone Bullet 272BH (about 5500 lbs.) and once we are loaded for the trip I'm right at the 14K limit.
I went shopping for my trailer with known capacities in hand. It fits us just fine but we spend most of our camping time outside the trailer. It is mainly used for sleeping and watching the tube.
The OP's challenge will not be that his E350 cannot tow up to 10,000 pounds, it will be the payload of the van. He has a fairly large family and will have a tongue weight with hitch of a little less than 1500 pounds. That does not leave a lot of payload capacity for other stuff.