He beats John Quincy Adams in a rematch in 1828 but by a much larger margin than in 1824.
Why would Adams be the anti-Jackson candidate and not Clay?
In 1832, Jackson's vice president John C. Calhoun loses to Henry Clay.
Calhoun wasn't "Jackson's Vice President". In OTL 1824, he was elected by a very large majority of the same electors who split four ways in Presidential voting.
The whole battle with the bank and nullification is somewhat butterflied out.
Not "butterflied". The Bank and Nullification disputes were driven by substantive causes that could not be removed by minor changes in circumstances (the flap of a butterfly). There might be substantive changes which pre-empt those disputes, but that's a
knock-on, which has to be worked out.