I would tell the doctor there is nothing in the statute about a current mental health exam. There is only a reference to your mental health history.
[§134-3.5] Disclosure for firearm permit and registration purposes. A health care provider or public health authority shall disclose health information, including protected health care information, relating to an individual's mental health history, to the appropriate county chief of police in response to a request for the information from the chief of police; provided that:
(1) The information shall be used only for the purpose of evaluating the individual's fitness to acquire or own a firearm; and
(2) The individual has signed a waiver permitting release of the health information for that purpose. [L 2001, c 252, §1]
Furthermore, I would instruct the doctor to honestly answer the HPD request based on your history on file and the doctor's current knowledge of your mental health.
Then ... wait. If the doctor actually follows through with his notice and prevents you from getting a permit, I would be in a lawyer's office that day. I can't even count how many rights and laws this would violate. The most obvious is infringing on your second amendment right. Also, he would be filing a false statement, saying you should not be issued a permit with no evidence or diagnosis to substantiate that claim. This one denial could prevent you from ever owning another firearm in Hawaii. The injuries this could do would be quite substantial.
If I were a lawyer, I would be all over this one after you get denied!!
Some people have all the luck!

I can see a pretty good payday coming to you because of this anti-gun doctor!
