the Catholic Church is their competition when it comes to running private schools and otherwise lucrative community support institutions
I generally agree with what you’ve written, but I think you’re assuming more pragmatism here than is actually present. Bitter hostility between Protestants and Catholics is as old as Protestantism (and much older than the institutions you mention).
Also, as a side note, there are plenty of Catholic Republicans. (37% vs 44% that identify as Democrats, according to Pew.)