Nah, he'd be supporting a Cruz'-Bloomberg ticket (sane)
A very different America around this time period as a result of Aaron Burr never being alive during the American Revolutionary era gives us three different parties: The Democratic Federalist party and the People's Republicans. Cruz, Bloomburg, and Burr would all be DF members. Bush, Clinton, and Obama would all be PR members. The DF, a bigger tent than the PR, are often paleoconservative but have a strong socially moderate and globalist wing like Bloomburg. The PR are very interventionist, think neoliberals and neocons. Opposing them both are the Community Party who are socialist, last led by President McGovern who died in office after pulling out from foreign affairs and just sitting back and watching the Holy Russian Front of New Rome collapse utterly in an incredibly bloody war. However when America was attacked by a terrorist splinter faction from the former HRFNR Hillary Clinton rallied the country to destroy the terrorists by invading Belorussia and, for some bizarre reason, Greece. (Controversial because the Greeks fought against the New Christian front in Belorussia
AND sent troops to back up American soldiers in Austria back in the 1960s!). She would win reelection pretty cleanly and be succeeded by George Pataki, who would be endorsed by Clinton in a tough primary against Obama. Pataki would get the blame for all the policy miscalculations of the Clinton years, only barely winning reelection against a strong DF ticket and having to deal with a DF dominated congress for his latter years in office. In '15 and '16 the DF would be confident of a victory but would face a tough primary because of a disunited party. Burr would be a force to be reckoned with during the early primary but a major scandal involving possible connections with midwestern terrorist group "Children of the West" would disrupt his campaign. The allegations would claim that Burr has been secretly undermining congressional efforts to enact legislation that would weaken the group's position in the country. The allegations are true but they have political undertones coming from Pataki's AG; they are still effective enough to weaken Burr in the state of Sesquwana and he would drop out and endorse Alberta senator Ted Cruz after only winning a few states like Vermont and Erie. Cruz would make it to a brokered convention who would make a successful gambit to unite the party by choosing his ideological rival, Michael Bloomburg. This unity ticket would crush Obama (PR) and Warren (Com) (Warren finishing second) and the Cruz presidency would start off on the right foot, having high approval ratings through 2018.
Burr's plan at this point is to run against Bloomburg in 2024 after cracks in the unity between him and Cruz develop then run against the two rival parties that are stuck in the past. But cracks reveal themselves quicker than thought in 2019 as Cruz and Bloomburg increasingly criticize one another. In 2020 things come to a head in a show of defiance when Bloomburg uses his tiebreaking vote in the senate to vote against the president's agenda to ban abortion. Cruz is devastated and congress loses confidence in President Cruz, who is forced to pick a new running mate in 2020 and falls against Joe Biden of the Pop Republicans. Burr would easily win the 2024 Democratic Federalist primaries but can he defeat an incumbent PR president far more popular than expected? It remains to be seen...