What about a "fake push" that does not leak message contents, sender etc.? Fuzz the time the push notification is sent by a random amount of time and you have something plausibly private given the constraints?

You're still dependent on Apple continuing to allow such a use.

If the goal is messaging that avoids government spying or censorship its a lost cause - the government would simply compel Apple to pull the app in their jurisdiction.

Briar is designed to work over 1) tor, 2) ad-hoc wifi, 3) bluetooth. None of those are going to be conducive to sending push notifications through Apple's servers.

That still exposes some metadata. Depending on your threat model, leaking the timing may or may not be a problem.

Also, how do you avoid leaking the sender? You can avoid giving Apple that information by routing the notification through a server, but then that server would know the sender and recipient.