I know that its not really going to help you get anywhere dealing with the PD, but the general statutes as I read them say that if they don't give you a permit application when you ask for one, that itself is an application. So maybe you already have the grounds to appeal to the BFPE?
My town also asked for things outside the state statute requirements, their instructions said to call and make an appointment to drop off the application, problem was the appointment maker never returned my phone call, in the end I just went there and dropped all the stuff off. Eventually the detective called me to and set up an appointment but that time waiting for the appointment was not included in the 8 weeks.
Sec. 29-28a. Application for permit. Notice of decision to applicant. (a) Requests for temporary state permits under section 29-28 shall be submitted to the chief of police, or, where there is no chief of police, to the warden of the borough or the first selectman of the town, as the case may be, on application forms prescribed by the Commissioner of Public Safety. Upon written request by any person for a temporary state permit not on a prescribed application form, or upon request by any person for such application form, the local authority shall supply such forms. When any such request is made in person at the office of the local authority, the local authority shall supply such application form immediately. When any such request is made in any other manner, the local authority shall supply such application form not later than one week after receiving such request. If such application form is not supplied within the time limits required by this section, the request therefor shall constitute a sufficient application. If any local authority fails to supply an application form upon the request of any person, such person may request an application form from the Commissioner of Public Safety or any barracks of the Division of State Police, and the time limits and procedures set forth in this section for handling requests for such forms shall be applicable.