Will it pull it, sure it will............but as mentioned it's far from the best setup. The 5.3 may not be the best but would get the job done, and is definitely not the limiting factor. You could put an 8.1L engine in the same vehicle and I still tell you it's not the best setup to use.
The suspension and overall durability are the issues. The suspension and tires used are made for ride quality and maybe carting a few extra adults and their luggage down the freeway, not hauling a big trailer. There will be a lot of stress on the rear 10-bolt axle and 4L60 trans.
If you can get a K5 on a trailer to only weight 7000 lbs., then that is pretty light. My 1-ton but otherwise stripped down K5 weighs around 6,000 lbs. and a trailer big enough to haul it will most likely weigh 2,000+ lbs.
A guy I know in our club pulls his big 10k deckover trailer with a 3/4 ton K5 on it to the trails every once in awhile with a 5.3 powered 1/2 ton 2wd truck (I'm sure this truck is much lighter than a typical Tahoe). It did okay but he did make the statement that "it was only a little hairy every once in awhile" (in regards to handling)........this statement scares me. My crewcab 2500HD pulled the same load along the same route with the cruise control on and one hand on the wheel with absolutely no "hairy" moments.