I'm just going to leave this here:
OP if forced to choose major-party candidates?
Explanation: Democrat, anti-racist, anti-papist. 
Actually I think I'd be a Hughes-Harding-Coolidge-Smith voter (Debs in 1920 and LaFollette in 1924 if third-parties allowed). Since anti-Catholicism in the 1920s came mostly from the Klan rather than Northeasterners, I would've firmly opposed it.
Actually I had no doubts your voting preferences would be exactly like that, but you know I like to trigger you.
By the way, what kind of voter would have feared Al Smith's Catholicism only because of possible ties to Mussolini and Primo de Rivera? It sounds absurd.
A liberal intellectual voter, as I said, who paid close attention to international affairs. It may sound absurd, but at that point the Catholic Church was still a profoundly illiberal institution in bed with reactionaries and dictators across the world. It wasn't just bigoted Southerners or the Klan who feared that a Catholic in power would bring in theocracy or take orders from the Pope, but secular Northern liberals too. In that spirit the National Liberal League had supported the Blaine Amendment some 50 years earlier. Here's a passage from The Puritan Origins of American Patriotism on just that:
The membership of this movement, largely well-to-do and well educated, came from the old Puritan regions in New England and from the areas of the Puritan diaspora in the upper Middle West, and they had important friends in Congress and even the White House. (President Ulysses S. Grant had made separation part of the Republican agenda.) The Liberals, or "total separationists," as I will call them to distinguish them from the earlier anti-Catholics, were in some respects heterogeneous. Some were atheists, some were Jews, others called themselves agnostics, and still others experimented with various forms of non-Christian "spiritualism" in vogue at the time. None of them saw any reason the United States should have any connection with Christianity, and they girded themselves to battle for "the absolute separation of church and state."
Liberals, then, rejected Christianity--but not Protestant religiosity.
[...]
At first glance, it is startling to see an unapolgetically anti-Christian movement flourish in a nation whose traditions and public institutions were steeped in Protestant Christianity. Yet a closer look would show that the Liberals were really located at the far end of a Protestant continuum [...] Even the Liberals, who disliked all forms of Christianity, could easily agree with the often-voiced Protestant view that Catholics did not think for themselves but took orders from a foreign power. In the campaign to pass the Blaine Amendment, Liberals formed a close working relationship with many pious Protestants. They were able to achieve this kind of working ecumenism because there was a broad Protestant consensus, at least in the North.
Also, somewhat tangentially I remember reading that in 1896 Mark Hanna courted Catholic voters as a winnable bloc for McKinley against the pietist Bryan because he saw the Catholic Church as a global force for conservatism. Furthermore that greatest of classical liberals, William Ewart Gladstone, strongly opposed Catholicism because of its innate conservatism.
I'd also like to address NC Yankee's point regarding the "Liberal Republicans." I get the sense that they called themselves "liberals" in order to distinguish themselves from the Radical Republicans then in charge ("radical" had an almost exclusively left-wing connotation at the time, by the way). In other words, they weren't trying to differentiate themselves from some sort of Republican "conservatism", but rather the radicalism that was then dominant in the party. You could say that their version of "liberal" meant moderate or compromising, a willingness to work with the other side. And yes, the other side was conservative.
You’re implying here that the Liberal Republican Party was some sort of conservative reaction against the “radical” Republican majority. I must say this is incredibly ignorant considering who founded and supported the party...
Carl Schurz: Founder of the Liberal Republican Party, 1848 Revolutionary and GOP Senator from Missouri
Charles Sumner, Abolitionist Republican Senator from Massachusetts
Charles Adams, Free Soiler turned Republican from Massachusetts. Later became a Democrat
Nathaniel P Banks: Abolitionist Democrat turned Republican, Governor of Massachusetts.
John Cochrane: Union General, War Democrat and Radical Democracy candidate from New York
Cassius Clay: Abolitionist Republican from Kentucky, later became a Democrat.
Lyman Trumbull: Democrat turned Republican, Senator from Illinois, later Populist party supporter who represented Eugene Debs and other Labour leaders before the Supreme Court.
Salmon P. Chase: Free Soiler and abolitionist turned Republican from Ohio. Later became a Democrat.
Horace Greeley: Editor of the NY Tribune, long time slavery opponent who often promoted radical ideas in his newspaper (He promoted Thoreau’s works for instance).
This solidifies that the formation of the Liberal Republican Party was rooted in genuine Liberal thought. Whilst it may be hard for present day individuals viewing the past through modern narratives to grasp, there was indeed great concern over the corruption and centralisation of power within the Grant administration, combined with disagreement over issues such as trade, and this motivated many Liberal minded Republicans to abandon it.
As for the two Conservative Parties you mentioned, obviously it was in the interest of the Southern elites to oppose reconstruction, however they obviously didn’t share the same motivations as the Liberal Republicans. Also, the VA party predates the Liberal Republicans whilst the SC party wasn’t formed until 2 years after the election so I’m not sure what you’re getting at here with regards to tying these parties and the LR’s together.
