There is no other way to send push notifications on iOS, you have to use APNS. When the app is active you can switch to your own local socket connection, but as soon as it goes into the background those connections are lost. Pushes can also start the app in the background if it hasn't been used in a while and has been evicted by the OS.

You can send push notifications with your own encryption on top, which I believe Signal does, so Apple can't see it on the APNS side, but your local extension to decrypt the content is still subject to the user's settings, and part of the notification history if you put message content in the notification.