I towed a 29' 5500# trailer with a 2011 Avy LTZ. I was usually over on payload by one or two hundred pounds as payload for my truck was 1,060# or something like that. I had the six speed trans and I swapped in a 3.73 gear. Power wasn't a problem in flat Florida but the towing experience wasn't the best. The suspension on the LTZ was a bit soft and the p-series tires didnt help either.
I think the trailer you referenced will be a bit much for that truck. I don't think you'll be happy with that combination. I bought a 2500HD and the difference is night and day.