As far as I know you do not need a doctors note if you are moving to the Big Island. I applied for my Handgun permit in early January and only signed documents and waited my 2 weeks for my permit. 
Nobody is required a doctors note, The release you sign allows them to contact your doctor to check to see if they believes you have a medical condition that would prohibit you from owning a firearm, their was a couple of insurance company's that would would respond with your medical records instead of answering the request with a yes or a no. HPD would not accept this and would deny permits untill you got a doctor to sign a letter stating you are fit to own.
My understanding of the law requires you to sign the form, It does not require you to provide your doctors information, If you have a doctor and you write None on the form you would be committing a crime, If you left it blank you should be good.
§134-2 (sub section C) :
(c) An applicant for a permit shall
sign a waiver at the time of application, allowing the chief of police of the county issuing the permit access to any records that have a bearing on the mental health of the applicant. The permit application form and the waiver form shall be prescribed by the attorney general and shall be uniform throughout the State.
[§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.