If I can follow your reasoning: you are thinking of the heat released when sulfuric acid is hydrated. You are right that it is quite exothermic initially, but as the solution gets more dilute, less energy is released. Therefore, you are looking for a way to use solar energy to re-concentrate the acid after it has been diluted to the point of uselessness.
You can distill water out of a solution of sulfuric acid until it is only 2% water, at this point it forms an azeotrope. Hypothetically, you can use solar energy to do this, but you really need to consider how much energy this process takes as well as how corrosive sulfuric acid is and balance that against the amount of energy you can actually "store" in the concentrated acid.
Even fairly dilute solutions of sulfuric acid are astonishingly corrosive, and a "battery" powered by the hydration of sulfuric acid would have to be made of some pretty special stuff. Especially considering what would happen if the battery was damaged and leaked. Honestly, I think there are better ways of storing energy harvested from the sun.