You can appeal the denial of your permit to the circuit court:
23-7-7.1. Requirements for issuance of temporary permit--Time requirement--Appeal of denial. A temporary permit to carry a concealed pistol shall be issued within five days of application to a person if the applicant:
(1) Is eighteen years of age or older;
(2) Has never pled guilty to, nolo contendere to, or been convicted of a felony or a crime of violence;
(3) Is not habitually in an intoxicated or drugged condition;
(4) Has no history of violence;
(5) Has not been found in the previous ten years to be a "danger to others" or a "danger to self" as defined in § 27A-1-1 or is not currently adjudged mentally incompetent;
(6) Has physically resided in and is a resident of the county where the application is being made for at least thirty days immediately preceding the date of the application;
(7) Has had no violations of chapter 23-7, 22-14, or 22-42 constituting a felony or misdemeanor in the five years preceding the date of application or is not currently charged under indictment or information for such an offense;
(8) Is a citizen or legal resident of the United States; and
(9) Is not a fugitive from justice.
A person denied a permit may appeal to the circuit court pursuant to chapter 1-26.
From § 27A-1-1:
(5) "Danger to self,"
(a) A reasonable expectation that the person will inflict serious physical injury upon himself or herself in the near future, due to a severe mental illness, as evidenced by the person's treatment history and the person's recent acts or omissions which constitute a danger of suicide or self-inflicted serious physical injury. Such acts may include a recently expressed threat if the threat is such that, if considered in the light of its context or in light of the person's recent previous acts or omissions, it is substantially supportive of an expectation that the threat will be carried out; or
(b) A reasonable expectation of danger of serious personal harm in the near future, due to a severe mental illness, as evidenced by the person's treatment history and the person's recent acts or omissions which demonstrate an inability to provide for some basic human needs such as food, clothing, shelter, essential medical care, or personal safety, or by arrests for criminal behavior which occur as a result of the worsening of the person's severe mental illness;