I’ve had this “problem” before, and I did not want to use the full Messenger app for privacy reasons as well. I had installed Messenger Lite, but it was discontinued a few weeks ago. I now have Facebook Lite, which also has Messenger, and you get notifications as well. It’s not perfect, but don’t give it too many permissions and you should be mostly fine. Using it in the browser is absolutely terrible, and you would not get push notifications.
Seems to still be available on F-Droid, but the last update was in 2017. I’ve installed it, but ended up not logging in since it’s targeting an SDK version so old that Android is giving me a bunch of warnings (and it doesn’t even scale to my entire screen, there’s a big black space at the bottom)