Looks like the SRT base payload is 1590, that's going to be really, really tight for a 600lb dry tongue weight trailer and a family of 4 or more. My trailer is 360lb dry tongue and loaded to travel and with 5 in the truck we were at ~90% of our 1650 payload. You're going to have to be very, very careful about how you load both the trailer and tow vehicle to keep all your weights within limits.
The RT appears to have an even lower payload rating so it's right out. If you're looking at SRTs make sure you check the yellow door sticker of the specific vehicle you're looking at.