I've seen a number of them sitting on 1-ton dually frames and similar things. For my COE project I have a P30 frame rated at just over 12,000 GVWR.
If my finished truck ends up weighing half that, I could reasonably haul a 6,000 load. That's enough for a 1/2-ton pickup and could even handle my 5,800 pound 1-ton four-wheel-drive. (Except I'm not building a car hauler.)
But if you look at the flat bed tow trucks out there, they tend to be bigger trucks than the 1-ton or 30/3000 series trucks like what I have.
Look for a bigger truck as a frame suspension donor and then measure and think. You might end up agreeing with Grigg or like me you may decide on a frame swap.
Keep posting your ideas. You'll get some help and lots of opinions here!