1) Minor nit-pick: It is a license not a permit.
2) Turn in only the state application. That is all they can require
3) Contact GO-NH/PG-NH
If they deny you (for not turning in their paperwork) SUE, and when you win, THEY pay your legal fees.
159:6 License to Carry. –
I. The selectmen of a town or the mayor or chief of police of a city or some full-time police officer designated by them respectively, upon application of any resident of such town or city, or the director of state police, or some person designated by such director, upon application of a nonresident, shall issue a license to such applicant authorizing the applicant to carry a loaded pistol or revolver in this state for not less than 4 years from the date of issue, if it appears that the applicant has good reason to fear injury to the applicant's person or property or has any proper purpose, and that the applicant is a suitable person to be licensed. Hunting, target shooting, or self-defense shall be considered a proper purpose. The license shall be valid for all allowable purposes regardless of the purpose for which it was originally issued. The license shall be in duplicate and shall bear the name, address, description, and signature of the licensee. The original shall be delivered to the licensee and the duplicate shall be preserved by the people issuing the same for 4 years. When required, license renewal shall take place within the month of the fourth anniversary of the license holder's date of birth following the date of issuance. The license shall be issued within 14 days after application, and, if such application is denied, the reason for such denial shall be stated in writing, the original of which such writing shall be delivered to the applicant, and a copy kept in the office of the person to whom the application was made. The fee for licenses issued to residents of the state shall be $10, which fee shall be for the use of the law enforcement department of the town or city granting said licenses; the fee for licenses granted to out-of-state residents shall be $100, which fee shall be for the use of the state. The director of state police is hereby authorized and directed to prepare forms for the licenses required under this chapter and forms for the application for such licenses and to supply the same to officials of the cities and towns authorized to issue the licenses. No other forms shall be used by officials of cities and towns. The cost of the forms shall be paid out of the fees received from nonresident licenses.
II. No photograph or fingerprint shall be required or used as a basis to grant, deny, or renew a license to carry for a resident or nonresident, unless requested by the applicant.