Please test these suggestions first in a discrete location!!!
I've been told that using nail polish remover, vineager, or acetone will work, depending on the type of material that the battle board is made from.
As for making sure that the GM can tell the difference, I would suggest that you put all the dry-erase markers in a special container that is only ever used for this purpose and take your sharpies and put them in a desk drawer or something. An ounce of prevention and all that...
Good luck cleaning your battle board!