I think you've got Beeper Mini mixed up with other iMessage bridges. The whole thing with Beeper Mini (vs other iMessage bridges) was that it was entirely client side on the phone, no server to block. So the "IPs Beeper Mini was using to connect to the APN service", those IPs were just the IP addresses of every individual phone with Beeper Mini installed on it, no centralized place to block.
No, the BPN server is a server side service that persistently recieves APNs to forward to the phone (that don't contain the message data) since unlike iPhones, Android phones can't persistently check for APNs (at least that's what I understood from the announcement article). AIUI that's what you're paying for. But that wouldn't explain why sending is broken.
The How It Works article is clear that BPNs is only used to serve push to your phone when the app isn't running. Disabling it would not cause send/receive failures.
If you check the How it Works post, they do show the Beeper Push Notification Service running in the cloud [1] to intercept 'new message available' APNs and then notify the Android device a new message is available.
