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Armor piercing rounds are not what you think they are. In 5.56 they will penetrate about 1/2 inch of steel which, while impressive, will only happen when the hit is direct. When you shoot something at an angle you are very likely to skip the round off.
For example, I have skipped 5.56 green tip rounds (they have a tungsten steel penetrator though they are not full tungsten, light armor piercing if you will) off car windshields when I was firing at the driver from an oblique angle. I have to note that this was part of a class at a range deliberately set up to demonstrate what does/does not work against vehicles and the people inside them. I have skipped rounds off car hoods, doors, drywall, etc.
If that same round won't penetrate 1/8" thick auto glass when it hits at an angle, what makes you think a much thicker bear skull would behave any differently? The odds of hitting that skull in the small region where the round would impact perpendicular to the plane of the skull and penetrate is very, very small as the area is probably about the size of a golf ball (or the eyes, of course).
Oh, and FYI, armor piercing rounds are some of the least lethal rounds on the battlefield. People who have the choice avoid them unless they expect to be fighting armored opponents.
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