I would also look at a 350. I have a '15 250 and we're tossing the idea around of a 5'er toy hauler also. I'm finding the limiting factor is the pin weight, not trailer weight. One of the trailers we're looking at is the Spartan 3210 with a dry weight around 11500 and empty pin weight of 2500-2600. I know once loaded with a 1000 lbs of Harley or atv's, the pin weight will drop since the garage is totally behind the axles. Per Ford, my max trailer weight is 15,100 with a 5'er, so I'm not worried about that. With a diesel in not worried about going, it would be stopping and overall stability.
A '15 F-350 SRW would only get me 600 extra pounds of trailer, but would leave more wiggle room for the pin weight with the higher rear axle rating. Don't know that it's worth the extra expense for me to buy a new truck. If the 5'er won't work, we'd probably keep our current trailer and wait on the class A or C that we plan to buy in 6-7 years or so.
My yellow door jamb sticker says 2150, even with the optional 10k GVWR package. My neighbor has the exact same truck in a 350, his says 3420 with an 11500 GVWR. My rear GAWR is 6100, his is 7000.
2015 Work n Play 30WLA
2015 Ford F-250 Lariat diesel, crew cab 4x4