Facebook Messaging

You can send anyone a message from your personal page through Facebook messenger whether you’re friends with them or not. But yeah, you can only send a private message to someone through your business page if they’ve messaged you first.
Could you not reply publicly from your business page saying ‘if you send me a quick private message I can reply’ most people know the Facebook rules- I’ve done that with a few people before and they’ve been happy to do it.