District court orders are absolutely binding on the states when directed against officials for violations of the Constitution, which this was. There is no credible argument that an order from a district court judge on a federal question isn't binding. Moore knows this and his only redoubt has been that the probate judges aren't really under the authority of the 'state' i.e. the executive and he has the sole authority to instruct them and that therefore no order is being violated.
He is going to lose.