It sounds like you have some back contact.

The instrument panel light uses very little electrical power. I don't think the change in alternator output will affect the light in the instrument because the battery is also connected to the car electrical system which can also provide electrical power. If the alternator has problem, the first thing you noticed should be the check battery light on from the panel after the engine starts. I would guess there is a back contact somewhere in the panel, and the problem gets worse when engine running at idle which migh cause some vibration in the car.

Good luck.