I can't directly answer about towing a 5th wheel of that weight with a 150, but I may have some data that might help.
First to the payload. To get max payload in a 150, I believe you have to go to the 18" wheels. With 17" wheels you lose 200-400lbs off the max depending on whether you are 2 or 4 wheel drive. You lose more with 4 wheel drive. That's a whole can of worms that you have to decide. I have an opinion and so does everybody else. We can cut to the chase and avoid the typical discussion and I'll just call you crazy whichever way you decide and we'll be done!
I tow a lot of trailers a lot of miles. Car hauler, boat, dump, cattle and RV. Everything but the RV I've towed with both a 150 max tow 4X4 SB and a 250 4X4 SB. With the boat and other bumper hitch trailers I didn't see a huge difference if any.
I do see a big difference in my gooseneck cattle trailer. I typically haul it at 7,000 to 11,000 pounds. Being a live load, there is some motion there even on a flat straight road. Pulling it with the 150, there's always a little steering involved correcting for the movement of the cattle and reaction to bumps. The cattle moving can change side to side loading and pin weight as you go. With the 250, you are barely aware there is any movement at all the stability on the road is much better. I would probably attribute it to both a stiffer rear suspension on the 250 and stiffer tires.
Whether that transfers to pulling a 5th wheel or not, I can't tell you. If I were pulling a 5th wheel of the weight you have, I'd go with a 250. My opinion is that it will give you an easier drive. YMMV
John & Kathy
2014 Grand Design Reflection 303RLS
2014 F250 SBCC 6.2L 3.73