I've been trying to figure this out and just end up being more confused.
One issue is that song's use of "ye" seems to be from the 1750s, so about 150 years after the KJV usage, which in turn was somewhat archaic. English changes.
> In Early Modern English, ye functioned as both an informal plural and formal singular second-person nominative pronoun. "Ye" is still commonly used as an informal plural in Hiberno‐English and Newfoundland English
So it's possible that the KJV uses "ye" only for formal singular second-person, even if was also used more widely used for informal plural, and as that informal plural became more widely used, it became the go-to way to translate the Latin, which is in plural.