and that is what the OP said: add a second middle name. Nothing about hyphens because technically, if you used a hyphen, it would not be adding a second middle name but simply changing the middle name. A hyphenated name is considered one name.
example:
Julia Louis-Dreyfus
first name: Julia
last name: Louis-Dreyfus
not sure how anybody can argue against it. Catholics have assumed a 4th name for a very long time with some of them using it formally. Never been an issue before.

