Special lighting for close-up camera

I have a game that uses a birds-eye view for most of the action, but uses small embedded cameras to show the action of characters when they do cool things. (You see the main camera and the close-up camera at the same time -- it just renders to a small window on the screen.)

The problem is these small "close-up" embedded cameras: they don't have good lighting, and I'm not sure how to fix it.

I tried adding lights to the scene, but they make the characters too bright from the birds-eye-view camera. I tried various Layer tricks, but ultimately decided that was a dead end, because the layer with the characters has to be visible on both the birds-eye and close-up camera. So if a light is shining on a character, they apparently "always" have that light on them... there's apparently no way to make a light "conditionally render" from separate cameras.

Is there a normal way to do this? Maybe I could brighten up the close-up cameras artificially somehow? Maybe something completely different?

Have you tried turning on the light (GameObject.enable) when you switch to the close up camera?