No one supports Apple Music. You can AirPlay it into a compatible device or you can use Apple Music from an android device and use Google Chromecast. AirPlay sounds better than Bluetooth and Chromecast is better yet.
Another option with Apple Music is to buy an Apple TV device and a monitor.. If you pick music from your mobile device and play it, then it will show up on AppleTV in the library and it will play at full resolution. If you merely navigate to the music on the Apple TV without first selecting it from the mobile OS, then you are limited to AAC, Apple’s version of MP3’s.
My advice would be to get a Wiim Streamer. Use the MacDac as another poster suggested. Try AirPlay with Apple on the Wiim (if you have an Android device then try Chromecast). I would also try a free trial with Qobuz which plays natively on Wiim and virtually every other streamer. Do some comparisons and then decide.
A Wiim streamer and a yearly Qobuz subscription should easily fit your budget. For that matter so would the AppleTV and monitor route.