My experience has been that there isn't a way to remove this sort of stain from cultured marble.

Try Bartenders Friend. I have successfully used it to remove stains from cultured marble.

I have had success getting stains out of cultured marble using beach-soaked tissues or paper towels. Soak the tissue with straight bleach and mash it onto the stain. Leave it there so the towel holds the bleach against the surface for several minutes or longer. I have even left them on over night.

Try mono sodium, It is a kind of seasoning widely used by the Chinese. I tried this one on my marble and it really works fine. You just have to dilute a teaspoon of mono sodium into a cup of water.

I'd try making a paste of oxygen bleach and water, apply and cover with plastic wrap and leave overnight.