1/2 ton towing a 5th wheel

Hello, thank you for all your responses. I am looking at a grand design Reflection fifth wheel-260RD. Specs are UVW 7,225. Hitch-1,325 GVWR-9,495. My F150 is rated at:GVWR-6,750 combined cargo wt and passengers-1,660 Rear GAWR 4,050 Front GAWR-3,300. Base weight-4,696 and it looks like from the Ford website, max conventional towing is 17,900 ( weight of truck and trailer together). I looked up what the hitch will weigh for my truck and it will be about 150 lbs. I don’t want to run out of payload and it looks like my payload is 2,054 (GVW minus curb weight) If I subtract the fifth wheel pin weight -1325 and the hitch weight-150, I’m left with 579 lbs for passengers plus whatever we put in the trailer. Which says to me, this won’t work. Would someone check my math and logic and let me know. I really appreciate all your help. So you know, we don’t currently have a fifth wheel so we cannot weigh it at this point but that would be the logical thing if we could.