Baked Reflection of Particles?

I need to bake a texture of an object. It looks great with hair. Baking the surface doesn’t work, which is not very surprising, but a mirror above the surface won’t bake a reflection of the hair either. Mirror textures work great for baking under other circumstances but particles aren’t showing up.

I have tried converting to mesh, then needed to extrude to make the paths into faces so it shows up on the mirror texture. The problem with this solution is with enough hair particles to cover the original mesh, Blender crashes when I try to extrude.

Is there any way to have particles show up on a baked texture, or on a mirror of the baked texture - without converting to mesh?

Or perhaps am I doing something wrong with the converted mesh, that it will show up in a render without extruding?

I have made an alternate texture that looks passable instead, but this is still something I’d like to be able to do. It seems unintentional that particles are not reflected when they render from a camera view but not in bakes - when everything else reflects in a bake.