In all honesty, I understand why people see it as a recurring problem for Paul - I've seen him do it in plenty of other threads - but I think you can make a cause for it working in this instance.
Dole has two successful terms, and his tax cuts provide the typical brief economic boom they usually do. After leaving office - under Kemp - the economy tanks as a result of them, much like irl, and Kemp's poor health proves a distraction from being able to do much to combat it and he is disallowed to run for a second term.
Like OTL, 2008 is an open field of candidates, but ultimately twelve years of Republican rule shifts back to Obama, with Gore and Kerry having already lost the same respective elections as OTL. Barack Obama, elected in 2004 in the fairly liberal state of Illinois regardless of president, jumps into a surely Democratic field and wins the election.
Obviously this scenario is fairly implausible - ignoring countless butterflies and basically re-synching with reality in 2008 - but it makes a lot more sense than some of Paul's other Obama-ending lists.