Who would theoretically vote for the Republican Party from Fremont through Trump

[Vice President Christian Man]

At least 2/3rds of the current GOP and likely higher but more specifically:

Socially conservative rich people
Northern Evangelical Christians (With the small possibility of Bryan)
Southern Unionists
Nationalists (Although possibly not in 1880)
Imperialists
Fiscal Conservatives (Although some might've gone with Cleveland)
Wealthy suburbanites would've fit through 2012, although I imagine some held their nose in 2016 before going towards Biden

[Vice President Christian Man]

Another common thread worth mentioning is opposition to immigration.

At least 2/3rds of the current GOP and likely higher but more specifically:

I agree with a lot of the groups you listed, but given how important the South is to today's Republican Party, I think 2/3 is a little high. Prior to the Civil War (and especially in 1856) Republicans had essentially no support in the South, and only minimal support in the border states. Those who would become Southern unionists mostly voted for Fillmore in 1856 and for Bell in 1860.

That is true, but demographics have changed dramatically since then. Had a rematch hypothetically taken place between then and now, I think there would be less differences than one would think.