I know Iceland has strict rules about naming, but as it’s also a socially progressive country, I’m wondering if there might be any allowances for non-binary people? A person who is neither a girl/woman nor a boy/man might not want to be referred to as anyone’s “dóttir” or “son”. Perhaps a suffix translating to “child” could be used instead?
And FWIW, speaking as a US-American, I think “Hawk” is a perfectly fine name. As is Jonathan. Choose whatever you like best, of course!