Flat bed will be much heavier and reduce available pin weight (unless aluminum). They are built to handle heavy loads placed anywhere on the bed, which is overkill for a fifth wheel with it's central load. Also the bed floor is higher, many flat beds use 4 inch long rails and 3 inch cross rails, so 6+ inches higher loading height.
My solution is different, a reduced height utility bed. The floor is about 2 inches higher than a stock pickup bed. Bed depth is 14", much less than a stock pickup bed, but top of the bed is just 50" on a 1 ton so I have about 10 inches clearance for the fiver. Just as important, I can stand flat footed and still operate the hitch. The utility bed only weighs 400 lbs more than the stock bed. Tailgate folds down full 180 degrees vs. 90 degrees for stock so it is never a clearance issue.
The custom utility bed wasn't cheap, it cost more than a simple flat bed but less than a hauler bed. But the bed is now 14 years old, on a second truck and will last longer than I will be towing.