Do you have any references at all for a winged stage being feasible in any way? Considering that there are plenty of rocket stages out there that can't even be moved horizontally unless they're pressurised I doubt anything vaguely resembling the Falcon 9 could ever be retrofitted in the manner you describe. What form would these, presumably retractable, transonic capable, necessarily light wings even take? How would they ever be as light as the remaining fuel load in an ASDS landing, especially when wheels are added?
Russians played with exactly such a stage, that had wheels from Mig-29 and engines from Yak-40. It was called Baikal and was feasible, but eventually never flew.
https://en.wikipedia.org/wiki/Baikal_(rocket_booster) Here is a Wiki article on it. Reverse: it was supposed to use engine from Mig-29 and wheels from Yak-40 :) Anyway, it never went anywhere beyond mockup stage.
