Breezie,
If you are convinced that you cannot do more damage to the back of the fouled anchor button, I can tell you how to take off the corrosion. Look in the sporting goods department of a store like Walmart. Buy a bottle of "Blue and Rust Remover" from the gun supply section. Apply it with a Q-tip, keep it wet with the solution, and let it sit or pour a small amount in a saucer and let the back soak in it. Every hour or so, take a small brass brush or sometimes a toothpick and scrap off what you can. If you can't see anything, repeat the process. In less than a day, it will be clean. Just go slow and stop when you see letters. If you continue, you may remove everything. It's not a nice way to clean things but it will get the corrosion off and if you are careful, you will be able to read what's underneath.
Another thing I have used for this type corrosion is to simmer it in Diet coke. Taking it out every hour or so and trying to use a toothpick to scrape off the crud. Sometimes that works like a charm and doesn't even harm the patina.
So if you are ready to take the road of no return on the back, start with the diet coke and if that doesn't do it, use the blue and rust remover. That WILL remove it.
Good luck.
Daryl