This is more feasible than people might realize. Ford switched parties to run for the Senate in 1918 at Wilson's behest and never looked back. Nor would he really be in the same lane as La Follette, whose support came from the proletariat and small farmers (the Socialist Party, farmer's groups, labor unions). Ford's base would be the middle class petite bourgeoisie- although the progressive muckraking of the turn-of-the-century had petered out into a dispassionate efficiency-fixated "business progressivism" following the horrors of industrialized total warfare, that was exactly what Ford was selling, and he was just the celebrity to get them excited again.
Of course, neither the urban Democratic machines nor the trusts would be happy, and they would run the mother of all smear campaigns. But then Ford could fire right back with Teapot Dome and punch from the right on cultural tensions with McAdoo's KKK connections (Indiana's a slam dunk). All in all, I think he pulls off an upset.
President Calvin Coolidge (R-MA) / Fmr. Budget Director Charles Dawes (R-IL)
Businessman Henry Ford (D-MI) / Fmr. Secretary of the Treasury William McAdoo (D-CA) ✓
Senator Robert La Follette (P-WI) / Senator Burton Wheeler (P-MT)