I have about the same configuration, an '03 5.3L/4x4 Tahoe, family of 4, 4200 lb 23' TT. It handles hills just fine. I've taken it to the rockies, coast mountains, cascades quite a bit It has the power. It runs reasonably cool, although I did get it a little hot a couple of weeks ago climbing the Siskyous pass in 104 degree heat.
The thing I don't like too much about it is that it has 3.73 gears and a 4 speed transmission, so it spends most of its time in 3rd and 2nd gear and there's a big jump between them so it's rather loud and clunky on a long grade. If you can swing one of the newer 6 speed transmissions it will be nicer to drive because there are more gears available at highway speeds. You don't need a 3/4 ton Suburban for a TT that size unless you are packing the SUV full of passengers.
I get 9.5-11 mpg towing with my Tahoe. I've put over 40,000 miles on it and the trailer and have had no mechanical issues other than normal maintenance like tire and brake pad wear. It's pushing 170k miles now, and still reliable, but I'm planning to replace it with a crew cab 1 ton truck next year because I want to get a 5th wheel.