The drive train on a school bus will have had a tough life starting and stopping and may be geared low since most operate on low speed roads. That low speed also allows for smaller engines which might not be appropriate power wise for Interstate use and heavy loads. A complete overhaul of the brake system would be appropriate. A lot more research would be required.
We started out with a new 2013 25FB International Serenity thinking our existing 2007 Mercedes ML320 CDI (turbocharged diesel) could do the job.
It was legal across the scales on the way home and handled the steep mountain East bound on I-10 out of Palm Springs by shifting down to 5th gear and maintaining 55 mph (the posted California speed limit when towing). After loading the trailer for camping and thus bringing the GVW up to 6,900 pounds of the allowed 7,300 pounds, we discovered with my wife and I and about forty pounds of stuff in the back of the car that the front axle weight as well as the GVW of the Mercedes was exceeded. The transmission was not happy with the heavier tailer load on basically level ground in the Phoenix area.
Thus we migrated to a new 2012 3/4 ton Ram 2500HD diesel which was more than adequate for the 25FB and it is still appropriate for our 31" Classic Airstream with a 10,000 pound GVW.
The published tongue weights in the Airstream literature are for a new bare tailer.
After installing a heavy hitch head (Over 100 pounds associated with the Pro-Pride assembly), full fresh water tank, 5 gallons in each of the black and gray water tank plus appropriate chemicals and all your kitchen stuff, clothes, kids must haves etc, one can find themselves with over a thousand pounds in tongue weight and not all of it can be off loaded to the trailer. One is exceptionally lucky to get even a third to the trailer axles.
My second son has seven kids. They are getting huge with each passing year. They use Ford 12 passenger vans to haul the family around and their useable payload is getting less and less each year as the kids get heavier.
The OP may find that if all seven of the OP's family cross the scales with a full tank of fuel and drinking water and snacks in the existing tow vehicle, there may little to no payload for a trailer left in the ratings.
Drive the Suburban to a truck stop and get the vehicle weighed when loaded with people, a full fuel tank and their "immediate access" stuff and compare the total weight and axle weights with the door label. Then you have the reality to work with instead of wishful thinking.
Then check the total Suburban scaled weight against the maximum towing combined weight of the car and trailer.
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TV - 2012 Dodge 2500 4x4 Cummins HO, automatic, Centramatics, Kelderman level ride airbag suspension, bed shell
2014 31' Classic model 30 twin beds, 50 amp service, 900 watt solar system, Centramatics, Dill TPMS, disc brakes, 16" tires & wheels