I have an equivalent weight camper with a tag dry weight of 3571. I've had mine 11 1/2 years.
I carried it to Montana and back ... approximately 1000 miles with no issues on a 2006 Ford SRW SC 6.0 PSD on the maiden voyage. I upgraded to 19.5s within the first 6 months to give my tires room for error and drove it 8 1/2 years that way. That upgrade in price was essentially the same as probably trading for a DRW. Where I lived and worked, a DRW was not an option since the SRW was my daily driver.
When I no longer needed a SRW, I changed to a F450 because it turns way tighter, has 19.5s stock, and has better brakes than either a SRW or DRW. No doubt the F450 with 14500 GVWR handles the same TC better than the F350 with 11400 GVWR even though the SRW is obviously way lighter.
But, I still miss the SRW if the TC isn't on the truck. The DRW mpg and range unloaded is horrible in comparison to my previous SRW. On the other hand, I had no issues towing my wife's uncles TT behind us with TC loaded on our trip to Yellowstone and was even under GVWR. I wouldn't have even considered that combination with the previous SRW.
Things to keep in mind.
1)Salesmen and the people here can't tell you what you should do.
2)SRWs are dramatically better in larger cities than DRWs.
3)DRWs are generally better hauling any TC although there are exceptions.
4)All DRWs can't haul any TC and still stay under the GVWR. Most of the people here with DRWs are over GVWR, too. DRWs will generally stop the load better than a SRW.
5)Not all DRWs are equal. The vary dramatically in capability. There are SRWs that I'd prefer over some DRWs for carrying a TC.
6)19.5s are generally always better than any other tire for strength but ride rougher. You can have equivalent carrying capacity of many DRWs if you upgrade to 19.5 Gs in larger sizes. My F450 with 19.5s rides rougher than a F350 with conventional Es.
7)Your biggest danger of being over your tire ratings is a blowout. Several here have experienced that, and it isn't fun and is expensive.
8)Being near the GVWR makes it hard to justify bringing all of your stuff if you want to pay attention to that number.
9)Many people here who are offering advice are really just telling you what they did. To many of them, the only perfect answer is if you agree with them.
Have fun making your choice and keep in mind that most people change their RVs and tow/carrying vehicle over time. If you don't need a SRW, you'll be ahead biting the bullet and trading for a DRW. You'll also save the money to outfit the truck with airbags, stableloads, tires, tiedowns, shocks, etc. You'll also more than likely will have enough capability for your "next" rv or to tow mostly whatever you want.
One of the best comments you'll see here is "shop for the TC first and then pick the truck"