I wasn't an an anarchist when I got married, but I was opposed to state interference in the institution. I got a government license because the pastor of the church we wanted to get married in required it.
I don't know how strongly you don't want the state involved, but I'd say to just go ahead and get the license, and call it notifying them like you're doing when you legally change her name.
Or maybe you can get the license made and use it to change her name really fast, but never actually file the license. There's usually a few days or weeks of grace to cover the honeymoon.