Sprite with 2 different textures on each side?

I’m using a Paper2D player in a 3D perspective top-down game. My character moves like this:

However I’m currently just putting 2 flipbooks 0.01 apart, each having its own animation state. Obviously this won’t work since the sprite overlaps where it’s not supposed to be. Is there a way to have one flipbook component have 2 Source Flipbooks, each on one side? If this isn’t possible, is it possible to at least turn off rendering on the backside of a flipbook?

Update: I made a masked lit material and turned Two Sided on; it works on normal sprites where it only renders the front. But it does not work on flipbooks; simply has no effect.