I would guess that instead of "slipping through the cracks" it was more likely "overlooked" or "ignored". And just ot look stupid, I'd say that the CSA states were never actually technically recognized as a separate country by the USA, so naturalization would not have been needed by the said children. However, the CSA states were "re-admitted" to the Union, so wouldn't that be "recognition"? In other words, too sticky an issue to deal with after such a terrible experience as the war, so "ignore or overlook it".