@Steve it's actually the way we've set it up right now.
We made a little script on our server which triggers a pushbullet whenever there's a new item in the recent.rss feed.
Although we've just run into a bug with the rss feed https://community.nodebb.org/topic/1419/bug-category-not-showin-up-in-recent-rss-feed#10477
We had to set it up locally, since our site is password protected over https, but if you have an open forum youo should be able to quickly set it up in on time via IFTTT, zapier or some other push service.
I think the next step will be to have also the personal notifications for each user to be sent straight via pushbullet, maybe via a script which matches the username with a pushbullet account.
We noticed that using the browser's notifications has its limit, since they only work when the app is open on the mobile device. Which kind of beats having the notifications altogether.