If you like your 2014 Suburban, I think you'd be most happy with a new or almost new (2010-2013) 2500 Suburban. It will have a lot more payload and be ready to handle any trailer up to at least 7500 lbs loaded. Any trailer over that size will quite possibly exceed a Suburban's (2007-current) hitch rating of 1000 lb maximum. The 2500 model should have enough payload for your family and 1000 lbs of tongue weight. They also get reasonable fuel economy with the 6.0L engine and 6-speed auto.
I own a 2003 2500 Suburban with the 8.1L. It would work well also but we use it as mostly just a tow vehicle since we only get about 11 mpg in day to day driving with it (unloaded). It is a great tow vehicle but gets horrible unloaded fuel economy. We get about 7.5 mpg towing 65 mph with our 7,000+ lb trailer (have not weighed it yet). Our 3 kids love all the room it offers. They can even ride 3 across the middle row on trips when we bring our bikes, we can fit all 5 bikes in the back with the 3rd seat folded forward against the 2nd row.
