Honestly, all of America's major political movements are at least slightly Classically Liberal. Even most Socialists in America support Freedom of Speech and Religion. America was largely founded on philosophy as opposed to history as Thatcher said, and Classical Liberalism largely was that philosophy.
Rockefeller Republicans aren't really Classical Liberals. They are moderate (relatively progressive) Republicans who tend to exist in heavily Democratic states to become electable (only way to steal a few seats in blue country consistently). They tend to support a hawkish foreign policy, large aspects of labor unions, the Welfare state (in general terms), Keynesian Economics, varying degrees of gun control, mass surveillance, the drug war, and other policies that can be difficult to square with Classical Liberalism (not to say these are entirely incompatible with Classical Liberalism[they aren't], but there is nothing particularly Classical Liberal about them compared to other factions of the Republican Party). Other than the the Trump populist wing, it is possibly the most authoritarian of the Republican Party factions. They could easily be called moderate conservatives who are slightly Social Liberals, but not really Classical Liberals.
A Party that ranges (typically) from Rand Paul to Gary Johnson, with a handful of extreme libertarians (like Ron Paul), along with people who are more conservative or progressive largely due to region (someone like Ted Cruz in Texas, Ben Nelson in Nebraska, Cory Booker in New Jersey [though they would have to change their minds and take the party line on a few issues).
If they ran Conservative Liberals (like Rand Paul) in the North East, they can probably compete in Maine and New Hampshire, but not elsewhere barring a scandal or something. If they run a more Cosmopolitan Classical Liberal (like Johnson), they could probably win New Hampshire and Maine most years while being very fairly close elsewhere. More realistically their candidates would range from a Gary Johnson type to a Cory Booker type, making them very closely matched with Sanders and Warren style populists.
