BAFT will supply you with all sorts of THEIR ID. Names, badge numbers, phone numbers, contact information, investigation case number.
I get suspicious any time someone asks me for information on the phone. They don't get my name, address, what I'm driving or who I bank with, where my mortgage is held -- if I hold a mortgage.
"Fishing expeditions" -- I think called "phishing." Asking for information so they can hack into accounts or whatever.
The BATF agent should be able to provide you with enough information about what they're up to that you feel secure in providing the information they ask for. It sounds like they're either looking to ID a stolen gun, or have recovered a gun and want to verify ownership.