How to login only to Google Music and not the other Google services?
As someone who tried to keep a select few Google apps from opening a bevy of other Google apps, just to preserve battery life, and that failed so spectacularly that I opted to use non-Google apps, I have a suspicion you really can't do it unless GMusic runs in kiosk mode (possible; I use iPods for kiosk music so I'm not sure about GMusic).

If GMusic doesn't have a kiosk mode, maybe you can run GMusic in a kiosk browser like SureFox? (I think you can, but it's been a while since I've tried GMusic). There are several of those to try.

But I imagine your best bet is seeing if one of the paid kiosk apps works well with your phone. Note that some of those apps reportedly "mess up" certain phones/tablets, so check the reviews on Playstore for your particular device before trying/buying one.