So: I use a custom resource pack, and it turns out that the animated parts of it don't work with Sodium. What I'm wondering is, could this mod be made to trick the resource pack textures, that usually use Minecraft's own core shaders, to function with Animatica instead when Sodium is installed?

No, it cannot
(core shaders reprogram the rendering pipeline to make animations work, there is no single way to just automatically convert that to animatica/optifine since it can be done in an infinite number of different ways)