Towing 6,000 with a 6,500 towing limit safe?

Tow ratings nowadays are defined by the SAE J2807 standard, which sets minimum acceleration times up a steep grade representative of Davis Dam. So yes it's safe, and the truck can handle it, but you'll be revving the engine high to make power. Shortened maintenance intervals also kick in, like rear diff oil replacement every 2 years or 20k.