Add thickness to hair path for render?

I’m trying to render this very dense amount of hair down onto a low res object for texturing (for a game, don’t want thousands of verts for a single object)

I want to bake its normals and diffuse to essentially a box around it, but the issue is, these are vertex paths so they don’t show up in bakes. I want to add thickness to them to make them cylindrical like hair, so I can bake them. How can I do that?