Had the milk paint been sealed with some type of overcoat (oil finish, wax, etc)? Or is it still a raw milk paint surface? Depending on how many coats of milk paint are already on the piece, you may have trouble blending a patch, particularly if overcoated (milk paint might not adhere). If you are using true milk paint, it may well require several coats to hide the sharpie mark. A single coat of milk paint is not opaque. Probably best to sand back and redo the whole thing.
One other point comes to mind -- is it actually *true* milk paint? Mixed from powder and water, with casein or milk derivatives as ingredients? Or is it something like General Finishes' "Milk Paint" which is actually NOT milk paint at all. The ingredients listed on the can are essentially acrylic resin -- no hint of casein or any milk derivatives listed among components. It is an acrylic paint, not milk paint.