Push Notifications don’t really exist for Lemmy yet, as they aren’t supported in the backend currently:
https://github.com/aeharding/voyager/issues/1027 https://github.com/LemmyNet/lemmy/issues/2631
As someone else here said, your best bet is probably monitoring your Inbox RSS feed.
If the main battery isn’t “meant to be replaced”, it will often act as the CMOS battery (e.g. MacBooks have been doing this since roughly 2008).