I can't seem to find the Effect Shader that turns you transparent when using the Invisibility spell in the list of effect shaders.

The only way I can find to access it is to make a magic effect of the archetype Invisibility, but I want to make an ability that turns the player invisible that doesn't actually apply any sneak benefits, so I was hoping to be able to use the effect shader itself.

Use a scripted magic effect.

Look at SetAlpha(), part of Actor Script.