At first, you look fine with the TV's payload and TVs towing capacity with that trailer, but get close to being overweight on the TV's GCWR. I don't know about the axle ratings. I got the figures for your TV here
TV's GCWR 12,000 lbs
less TV's curb weight of 5,474 lbs
= 6,526 lbs remaining for the TT.
The TT's GVWR = 6,691, so you would need to be careful of loading up the trailer and TV. However, that doesn't account for you, your wife and the dogs in the Suburban.
A better way to look at this is to add the passenger weight and dogs to the TV; then your truck's weight loaded becomes at least
6,074lbs (could be more so you might want to weight the vehicle).
So, with passengers and cargo in the truck, the GCWR of 12,000 - 6,074 leaves you with 5,926 that you could tow and stay within Chevy's specs for towing. That's a fully loaded trailer weight, too. You will load the TT with supplies and mods and propane, water, etc. The actual trailer that you look at when you're at the dealer needs to be less than this amount.
See what I mean by doing the math yourself? You'd think that a Sub should be able to handle it (at first glance), but it looks like the trailer is a little too heavy for your TV.