Unless the local cops object to your having such a license, the BATFE checks you out, if all is well, you get your license in 30 days or so.
It's important to point out that local law enforcement only has to be notified
that you've applied for a C&R license- you do not have to wait for them to approve
the license. It's not comparable to the approval that's required when purchasing certain NFA firearms.
Draft a formal but short letter with your name and address, and state that you're required by federal regulations to notify local law enforcement that you're applying for a federal firearm collector's license. Address it directly to your local police chief, sheriff, or whomever, using the default address from their website or the telephone book; don't call and ask where to send it, you'll just confuse them (that's the voice of experience speaking
). State that the letter is intended solely as a notification, and that they are not required to take any action. Send it certified mail, retain a copy of the letter and the mail receipt, put them in a secure file, and forget about it.
In most cases, the letter presumably gets stashed in a dusty file box somewhere, or is thrown away.