Removing 4x4" wood from base?
As you have access to the bottom of the post, consider to use a wood bit and drill a series of large diameter (1") as close to the perimeter as possible without striking the metal. Go as deep as the metal bracket, which will weaken the wall of the post.

Obviously if you intend to re-use the post, it will be less well suited at the full length. As such, you can also cut off the post above the bracket and perform the drilling all the way through, creating an even weaker bond with the bracket.

Inertial is important in this type of removal. If you can cut a hole in a sturdy plank that is exactly the size of the post, but not as large as the bracket, brace the plank and drive the post through the hole with it's own weight. When the bracket reaches the hole, it will budge. It may take a few shots to break it free.