How do I get an FAA okay for a private runway?
You're probably already aware that the FAA itself does not require the operator of a private airfield to receive any kind of approval from the FAA. However, there is a process in place to register a private airfield, and indeed the notice (not approval) is mandatory. The result of that process might satisfy your local authority.

The three FAQs starting here are pertinent. You need to file FAA Form 7480-1, either by mail or via the OE / AAA system.