Getting camera to show objects in a layer without the lightings

I have a `terrain` and a light callled `Light A` in my scene. `Light A` shines on the terrain. `Light A` is also in the `Default` layer, while the `terrain` is in a `Terrain` layer. `Light A`'s culling mask is set to the `Terrain` layer. This shows up fine on the `Main Camera`, whose culling mask is set to `Everything`.

I have another `Map Camera` in the scene that is also looking at the `terrain`. This `Map Camera` is suppose to show an overview of the terrain in the scene. This camera's culling mask is set to `Terrain` layer.

However, the `Map Camera` seems to be showing the `terrain` with the light shone on it from `Light A`, which is in the `Default` layer.

Ideally, I want the `Map Camera` to capture the `terrain` without `Light A`. I want the `Map Camera` to instead, receive light from `Light B`, which has a different set of configuration more suitable for a minimap.

So how can I get the `Map Camera` to cull mask the `terrain` without light shone from `Light A` on it, but instead with `Light B` shone on it?