Making a new post here because it’s easier than replying to everybody who brought up the same points.
Thanks for the input from everybody. I can’t give you the reason why my characters want a second baptism performed, as it would be too much of a plot spoiler, but the short form is that this is a fantasy novel; the reason for it is metaphysical, and can’t be solved by that character going to confession. You’ve given me what I need, though: the reason why baptism isn’t repeated, and then the conditional form that the priest would use once they convince him, against his better judgment, to do it. (I suppose I could have the characters perform the baptism themselves, but it loses a bit of that ritual and narrative oomph, which they would be very eager to have on their sides.)
So now I can write an argument about whether there’s any other power in the world capable of annulling the gift of God’s grace, and that was what I needed. Once I have the scene written, I’ll find somebody who knows the specifical historical Church practices to read it over for me and tell whether it works. (If any such person reads this post, do let me know; the scene takes place in 1884, post-Vatican I but pre-Vatican II.)