Is it possible to bake a normal map from another object together with its diffuse(accurately)?

Is it possible to bake this smiley along with its diffuse color onto the object behind it, so that the colors are relatively accurately applied on the baked on normals? I know how to bake just normals, and I know how to bake diffuse in a projected way, but it never applies correctly at all.

The reason I don’t want to necessarily color it after its been baked is because sometimes it’s easier to color when you can hide geometry and use the topology to color specific things.

Please refer to the video

You can adjust the extrusion.

https://docs.blender.org/manual/en/latest/render/cycles/baking.html