I do not think you can hunt in Bremerton, although you can hunt in unit 627. Kitsap county has plenty of public land. I hunt in North Kitsap county. If you live on NSB Bangor, I can only think you are active-duty. In which case, you can hunt on NSB Bangor.
mgdavis suggested Tahyua . . great around the tree-farms and Vashon Is. I know this post is real late, accept it may help someone.
Happiness is a warm gut-pile
on a cold day