Background refresh being off doesn't help for the Music app - it's one of those apps that run in the background anyway!
I did a few things that seemed to have helped and I'm no longer seeing the Music App show up in battery usage. I don't know for a fact that this is what made the difference or if it was just coincidental. I'd be interested to know if others try this and find that the Music App no longer draws down the battery.
Settings > Music > Turn off "Show Apple Music" (Note - I never joined Apple Music so I'm seeing a "Join Apple Music" Link. If you've already joined Apple Music, you might want to cancel to see if that helps. I also went to iTunes on my Mac (uses the same Apple ID) and made sure I wasn't linked to the Apple Music subscription service (since I used to use iTunes Radio) by logging out - did this on all devices that use my Apple ID.
Also,on the iPhone:
Settings > General > Restrictions > Turned off "iTunes Store", "Apple Music Connect" (and iBooks Store and Podcasts for good measure).
I also made sure that the Music App was turned off under Settings > Cellular > "Use Cellular Data for" I didn't want this background activity to burn up Cellular Data as well as battery!!
If it does, indeed, turn out that this was caused by the new Apple Music subscription service trying to connect in the background and wasting the battery (and using Cell Data), I would be pretty upset. It seems like Apple is more and more rolling out this stuff that, if you don't want it, you have to hunt down in settings a way to turn it off. If it was just coincidental, I'm still not happy given that the music app was somehow "enabled" in the background even though I wasn't intentionally using it.
I hope this helps.
Art