Law Enforcement Notification. The applicant must provide a copy of the Form 1 to the chief law enforcement officer (CLEO) who has jurisdiction over the area of the applicant's address shown in item 3b of the Form 1. In addition, if the applicant is other than an individual, a copy of the Form 5320.23, National Firearms Act (NFA) Responsible Person Questionnaire,for each responsible person must be provided to their respective chief law enforcement officer. The chief law enforcement officer is considered to be the Chief of Police; the Sheriff; the Head of the State Police; or a State or local district attorney or prosecutor..
So, I think you could get away with sending it to FDLE. But???
There are other ATF documents that address the specific issue, but technically I think this is the latest.