There was a lot of prejudice against Catholics in 19th century America, especially because most Catholics were Irish immigrants. In the decades following the American Civil War there was a lot of Italian immigration that was the target of similar prejudice. Anti-Catholic prejudice remained strong into the mid-20th century (we can't discuss it here, but I recommend looking into 1920s politics for more on this issue).
Catholics were a particular target of the Know Nothings (American Party) of the 1850s.
Besides the correlation with certain ethnic groups that were the target of prejudice, the loyalty of Catholics was questioned because of the Pope.
Here's a starting point on the subject:
en.wikipedia.org
I don't know that any particular Protestant denomination was dominant, North or South, in the 1860s.
That's not to say it was impossible to successful as a non-WASP.
By way of comparison: Ely Parker was on Grant's staff and became a Brevet Brigadier General yet I hope nobody would think that somehow disproves widespread prejudice against Native Americans in the 19th century.