Google Books has a lot of older Catholic military history books.
A lot of the Irish and German units were all or mostly Catholic, so that’s a pretty easy place to start looking.
You can also search for things like “Catholic chaplain” “Civil War”, and find stuff that way.
You’ll also see stuff about how some of the Southern bishops kinda started to slide toward teaching that Catholicism permitted slavery, while others refused to teach that, and how the Pope had to send all the American bishops a letter saying that slavery isn’t okay; and about the surprisingly sympathetic letter the Pope later sent Jefferson Davis after he lost big, because that’s the sort of thing popes do.
You’ll also see some interesting stuff that nuns did during the war. (A lot of nursing, for one. It made a big difference to how Protestant Americans thought of Catholics.)
It’s a good topic for research, with lots of surprises. There are many proud and shameful moments, and you’ll learn a lot about human nature.