From my experience the scrobbler only work with music I own in Apple Music, not any music I'm listening to from Beats Radio or any of the recommended playlists.

On the Mac http://micropixels.pl/neptunes/ works just fine also for streamed music from apple music not just the tracks you download into your local library. Not aware of anything that does it on iOS though unfortunately :(

On macOS, http://bowtieapp.com is a good free alternative.

Are there any scrobblers that work with the iOS version of Apple Music? I tried using QuietScrob, but that requires every track to be in your "library" to work.

been using last.fm with Apple Music since the beginning.

How? Nothing I tried worked with streamed music, only stuff I had locally (especially on iOS)