That's a lot of abstraction to implement a simple update check and I suspect it's very much not worth it to save a minuscule amount of battery life on a laptop. This is ignoring that you're incorrect about how Web Push works, so you'd need that extra TCP connection anyway and at that point, there's no point in bothering with Web Push. FCM is the same deal (and now you get to pay for extra stuff for a trivial problem, woo) and APN seems like the wrong solution to auto-updates.

Just poll every launch or 24 hours and move on.