I used to own a 2001 Burb 2500, same engine and rear end as yours. I towed a TT of similar specs to yours as well. I did not run any long trips out west or anything...mostly weekend trips around TN, but did go over Monteagle which is a 6% grade. It handled it very well. Not a lot of excess power, but sufficient. If I could have found a nice 8.1 I would have loved it, but the 6.0 did the job.
One thing I did do was replace the stock hitch based on issues some people have had with the OEM hitches. I did not have problems with my hitch, and some may say I wasted a couple hundred bucks replacing it...but it was worth it to me for peace of mind.
