This Bush would have reached the eligibility age of 35 for President in 1831. It's doubtful he could have stopped the Andrew Jackson reelection steamroller in the 1832 election and probably not Jackson's choice of successor, Martin Van Buren, in the 1836 election.
William Henry Harrison ("Tippecanoe and Tyler Too") was the landslide winner of the 1840 Presidential race and it's doubtful this Bush could have won that election.
In 1844, annexation of slaveholding Texas was the big issue and virtually unknown "dark horse candidate" Polk defeated the very famous Henry Clay who was somewhat ambivalent about the Texas issue. How much more so would an unknown anti-slavery Bush have been beaten by Polk?
In 1848, another war hero, Taylor won the election so, practically speaking this leaves only 1852 and 1856 for a 19th century Bush Presidency. Both those years elected one-term Presidents, known as "doughfaces" (Northerners who were sympathetic to the slave-holding South) who won only because they were guaranteed not to rock the boat on the slavery issue, something Bush no doubt would have done given his strong anti-slavery beliefs.
The fact that the Presidents elected in 1852 and 1856 were virtual unknowns shows that an unknown like Bush could have been elected but only if he had the "right" views, which he did not.
Aside from a significant POD,which I cannot begin to guess at, this seems an insurmountable "challenge" to me, Fabilius. Thanks for letting me learn about this fascinating character.