Pick him up, and pull it out
To make this easier, you can put him into a pillow case, with only a bit of the upper part of him sticking out. Keep the rest of him "balled up" in the pillow case.
Then GENTLY but firmly grab a hold behind the head and pull out the bark. If he struggles badly, kind of "go with" his movements, but don't let go.
Have tweezers and another person there, if that helps.