I've the same issue. The apple music is working with casting chromecast.
I've spoken with the apple support, this was realt disappointing. They don't how the app is working. I had to explain everything. At the end they it is a problem of an other supplier, while it is really the app from apple. I asked to escalate, this was impossible he said. I said escalate to your boss, or to the development department of the apple classic music app. At the end he said he will call me back next week. When there is a solution I'll let you know.