How can I repair battery acid damage on a Caesar-stone counter?
I would experiment with dye on a cut-off piece..  talk to some installers of ceasar stone or get some samples.

The dye I'm speaking of is a metallic salt type (Transtint http://www.homesteadfinishingproducts.com/htdocs/TransTint.htm ) They are intended for wood, but will stain most any material with some porosity to it.  

This will be a temporary fix, a surface treatment that will have to be renewed periodically.  If you get a mix working (keep track of your formula), you can bottle it for refreshing the color as needed.

I'd suggest 1 OZ of denatured alcohol and 4-6 drops of a Transtint color to start.  Mixing is straight forward and color blend is the true genius of this system.
PS wear latex or nitrile gloves..