I am trying to get my girlfriend legally deemed unfit for my children and take full custody. Without her, I cannot afford where I am living right now so I would be moving out. Is trying to protect my children in that manner and moving out a viable legal reason to break my lease? Im not familiar with Washington state real-estate law.

I'm sorry to say, probably not. Check Washington's Landlord Tenant law to be sure. Here's the link: Title 59 RCW: Landlord and tenant

Speak honestly and humbly with the landlord to see if you can get out of the lease.