There's a website I use that sends me notifications using Web Push. So I'm receiving these notifications just fine on macOS via Chrome. I'd like to forward these notifications to an iOS device via Pushbullet but the website doesn't provide a way to do so.
So I solved it right now by reading macOS' notifications sqlite database and sending the ones I want to Pushbullet's API and I receive them on my phone. I was wondering if there's any better way.
I tried opening the Chrome DevTools and recording Background Services → Push Messaging and I can see a log of notifications in json. How can I read those notifications?
This may not be the answer that you're looking for, but have you considered installing Google Chrome on your smartphone and registering that device with the website for push notifications? As pointed out in the comment, Chrome on iOS does not support Web Push.
Another suggestion: while creating the push subscription the website also registers a service worker. This service worker responds to incoming push events. If you open Chrome DevTools, go to to the Sources tab, and open the service worker, you can add whatever code you want. I'm not sure if this change persists when you exit Chrome though.