You'll definitely need a 3500 for that, likely a DRW. I've always considered a SRW to be good up to about 15K loaded weight. That 14K dry trailer will likely weigh around 16-17K loaded. That's roughly 3500-4000 on the pin which is nearly all the available payload for a SRW. Add in a hitch in the truck, people and gear and you'll be way over.
You "might" be able to technically stay in the numbers if you don't load the trailer too much and travel with barely anything in the truck but that's really not a fun way to travel. All that said, just the hitch in the bed and the trailer loaded with minimal stuff could put you over your payload. I'm trying to remember but it seems like most SRW 3500 diesels have payloads around 4000-4500#.