So, I will say that disabling the "[CS] Extra Characters" mod by Coop Deluxe Team helped get the custom voices working on my end. Previously, it would only play Mario's voice clips or not play anything when performing different actions. Now, it plays all of the characters' voices perfectly. The only thing of note is that it will still show a script error message when performing an action with some characters after loading them from the character select menu (I've seen it happen with Vivian, Princess, Piper, and Lakilester so far). It does not show anymore after they are loaded in for the first time, even after switching to another character.
Funnily enough, this also fixed the voice issues I was having with other character mods (the Kiryu mod would play clips from Mario and Birdo (from the extra characters mod) for actions not in the custom moveset for whatever reason).
Hope this is helpful.
(P.S I am using the OpenGL release of sm64coopdx. No idea if this works for DirectX)