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Hi there!
Yes its possible to integrate to 3rd party push notification system using better_messages_send_pushs php hook, here you can find the example:
https://plugins.trac.projectdmc.org/browser/bp-better-messages/trunk/addons/onesignal.php#L23
But that hook will work only in WebSocket version.
For the AJAX version you can use other hook to catch sent messages and then process your custom code to send push notification in your application:
https://www.better-messages.com/docs/development/php-actions/better_messages_message_sent
Thanks!