Greetings and welcome!
Short answer is you need to look at trailers that are in the 5000lb dry range (around 6200 loaded). Any bigger and you are going to exceed limits starting with payload.
Here is the long answer
The weak point for all half ton trucks and SUVs is payload. It has not been a power issue in a long time. Payload as you saw on the yellow sticker is the maximum allowable cargo and passengers. It can vary a lot but most half ton are in the 1200-1700 range. This is the vehicle's GVWR minus it's curb weigh as equipped from the factory. All people and stuff subtract from that.
That wonderful maximum tow rating of 9000lbs is for a specific type of trailer, a flat deck or boat trailer, which has a tongue weight of 10% the total. The real hard number is the hitch rating. 900lbs with weight distribution. if the tongue weight is 10%, then 900/0.10 = 9000lbs!! See how that works? Problem with RV trailers is they have heavier tongues in order to keep them stable, like 13-15%. So with that same hard hitch number of 900, 900/0.14 = 6429lb!!! WOAH big difference huh? If you hooked up a 9000lb RV trailer with 14% tongue weight, the tongue would weigh in at 1260lbs blowing past that receiver rating! But neither the car dealewr or RV dealer will tell you this!
Now lets look at payload. You can only use that full 900lbs hitch rating IF your total of people and stuff is less than the diffence. So 1547-900 = 647lbs of people and stuff. With a family of 7 that is not going to be easy. Maybe now but as the kids grow you will get heavier.
So total up all the people, carseats, diaperbags, strollers etc and subtract from 1547 and this is what you have for hitch weight. Lets say you are around 700lbs for 2 adults 5 kids carseats etc, that leaves 800lbs for hitch. Then divide by 0.14 to get the max loaded trailer you should look at. 800/0.14 = 6050lbs.
A good general guesstimate for all the stuff people add to a "dry" trailer is 1200-1500lbs. This includes batterie, propane, food, stuff, pots, pans, any options, water, televisions. etc. So pack light and take 6050-1200lbs = 4750lbs dry weight. Or lighter and look for 5000lbs dry. This will get you close to your max numbers but will work. Heavier and you are going to have issues. I have loaded my Expy with ceramic tile to it's limit, it was not too happy. You can find family trailers in the weight range but your options will be limited. As an asside, my 22ft TT weighs 5200lbs full loaded ready to go. It sleeps 7.
The EL will handle longer trailers ok, your main issue is going to be payload. Half ton SUVs were ment to haul lots of people or lots of stuff, but not necessarily both at the same time.
Good luck!
