Is there any way that I can send a signal from chests inside a factory to the overlay controller so it shows an item count on the overlay?

While theoretically possible, this is not the purpose of the overlay. To conserve UPS, the overlay is only updated whenever a player leaves the factory building. As an alternative, you can send an item count out of the factory building yourself, with circuit connections, then display it there using lights and circuits, or some other modded signal display tool.