Id do it with out thinking twice with my 2000 c2500, and it has the vortec 350 also. I pulled my old 24' TT that usually was in the 6500 lb range a couple of times, no issues. I've pulled my equipment trailers upwards of 8K with out issues.....
You probably have the 1500 version. You can tow with 3.42 gears/4l60e trans and have better performance than a 2500 and the 4.10/4l80 trans. As the 4l80 has a 2.48 first, the 60e a 3.08. Overall low with the 60E is 10.55-1, 80e a 10.1-1. The engine with rev, so plan on it reving on hills in the 4000 range. It develops 255hp, 330 lb ft of torque. you'll do fine if you keep your head togother, and not use your right foot for everything.
You mihgt change your gears to 3.73 if that is not what you already have.
Marty