You can do this without a hybrid app, native app, or requiring users to install Pushover. It is possible to build a “progressive web app” that can send push notifications even after the page has been closed.
These work today in Chrome for Android (and other browsers seem to be implementing support), but there is no support on iOS.
Check out this demo, and this guide explains how to do it: https://developers.google.com/web/fundamentals/engage-and-retain/push-notifications/