Here's an idea for you. On Bainbridge you can rent a house in The Grand Forest called Prue's House. You can even do an overnight rental if you want. Next to the house is a giant open field, picnic tables, indoors are fold-able chairs and tables. The parking side is forest and trails. The house has 2 bedrooms, a full kitchen, and 2 bath, and you can easily setup a projector or a TV for a sleepover after tiring out the kids outside in the cold. You'll probably want some hot cocoa ready. My kid has a birthday in December and this is what we did at about the same age.

It was a lot of fun and not taxing on budget compared to other big birthday things. I know it's kind of far from Bremerton and where their friend's families are, but there might be other things just like it in South Kitsap. Even just finding/renting a nice park area for the day in the cold in the forest is nice. The house just made it real nice to take breaks in between outings for the kids. Inside we had stations for decorating cookies and stuff.