The inner Lagrange point is always between the sun and the earth (by definition), so the "shadow" would always be centered on local "noon" and would pass the each longitude at the same time every day.
But the sun isn't a single point, it is big! In the same way that a pencil can have a sharp shadow but an office building will have a blurry shadow, the satellites would produce blurry shadows.
I'm not sure of the dimensions of the inner Lagrange point-- I agree that the shadow would be centered on astronomical noon, but I think it would be so blurry, and have such a wide blur radius, as to be unnoticeable except as a change in absolute light.
