After some investigation it is not that straightforward to implement this, because iOS app needs to be built with distribution type certificates for the push notifications to work on browserstack devices. Because of the distribution type, webview is not accessible via appium. Without access to webview, it would be needed to do the test only via UI, which is again not that straightforward because we could only use native selectors. If I remember correctly only xpath selectors could be used with distribution type hybrid app, which are not stable, as they can change in different devices/os versions.

Closing this issue as won't do for now.