It would've taken a minor miracle to get Bumpers to run for President in any election. They courted him in 84, he declined. He flirted in '88 (probably his best chance), but stayed out, and by '92, he seemed completely unwilling to run.
A Bumpers Presidency probably would look a bit like a Clinton one, minus the slick imaging and scandals. Bumpers was a moderate whose charisma was very folksy. As President he probably would be similar in temperament to the old clerk taking care of the general store. Bumpers had a way of smoothing things over that the Clinton White House never seemed to do.