Mix up the bleach and apply a test bit on the dark edges of the stain to see if it will work.

If it does, apply to the dark edges and when they are almost bleached out, do the entire top for a very short time to equalize the color.

Rinse, neutralize (whatever the container says to do), then let it dry and sand lightly.

The article linked explains the various wood bleaches.

Oxalic acid is what you use to remove stains caused by tannin reacting with iron. I"m not sure that's what you have. Common to see black spots on woods like oak that have been wet and are in contact with iron or steel.

I'd probably start with A-B bleach and see how it does, and yes, do the whole top.

Here is a link that might be useful: http://www.antiquerestorers.com/Articles/jeff/using_wood_bleach.htm