I'm not really a friend of being disturbed by notifications. My #Librem5 does only play sounds on SMS. Anyway...
What I've been thinking of especially since there is some work on the linux system of the #Pinephones Modem is if there wouldn't be a way to implement some service into the modem that does work even when the phone is in suspend mode.
This wouldn't solve the quest of having some service providint push notification, but it would solve the problem first that on the Librem5 and the Pinephone push notifications can't be received while the phone is suspended.
Another way to get this without having to hack on the modem would be to run a service that would actually voice call a phone when it's supposed to get a notification to wake it from suspend. This solution could be illegal in some jurisdication, because you signal something by a call that is not intended to be taken (and thereby charged).
After a wake-up from suspend by whatever dedicated method the linux system could do whatever would be good to get information about the event that initiated the wake-up call/[wake on modem data thingy].
From there whatever happens in linux could evolve. At first on the phone side I thought one would wait for the services defined by the user to connect to their servers and exchange informations. But this could also evolve into some protocol where the phone asks at the wake-up service for the reason for the wake-up and just make sure that it acts upon that information.
Maybe I misunderstood the whole situation and the whole problem, but it's nice that you started the discussion about this topic and I'd love to learn more about other ideas.