Barry, by his choice, will run for office. If he chooses to be placed on the ballot in AZ, he will have to conform to their state law. He is free to choose not to be placed on their ballot.
You are correct that running for office is not a right, IMHO, but the courts may see it as an equal protection/equal opportunity issue.
If Obi is denied a ballot position, he will sue. Then the courts will rule that the Hawaii BC is binding on AZ and every other state under Article IV.
AZ courts will be over-rulled by the Feds. Some idiot lawyer will argue that birth certificates issued under state seal are the same as driver’s licenses. Or that the Passport Office recognizes all BCs equally from every state.