This is just a concern I had with the client. Back during the good ol’ Reddit days, I get notified about a post/comment reply immediately when it happens. I have since left that website, and for some reason I don’t know why this isn’t a feature yet.
And yes I have tried Connect for Lemmy, I had major problems with that app.
Yeah it’s a json rest API so you could easily write a script to poll for new messages. This is unfortunately the most authoritative implementation at the moment: https://github.com/LemmyNet/lemmy-js-client