How to get rid of any animal in the world.

1. Shoot it.
2. Trap it.

If you can't shoot it or trap it, remove it's food source.

In your case the food source is bees.  The bees are also doing damage to the structure of your home, you may not be able to see it with your eyes, but they'll destroy those beams from the inside, whereas the woodpecker starts works from the outside.

You might consider it a blessing that the woodpecker is telling you that you have a bee problem.