I am likewise having the same problem with my Lumia 640XL with version 1511, build 10.0.10586.218 (planning to update to 10.0.10586.318 tonight, we'll see if it helps..) I haven't had any issues with the player other than the bluetooth, and I'm ecstatic about it's existence since Groove is really a horrible program on Win10M!
The GUI is excellent, it's fast and responsive, and has a lot of control and options for ensuring how it is supposed to function!
One of the things I'd like to see (and maybe already exists?) is the ability to create a playlist on my Win10 desktop and have it sync? That would definitely be sweet. However, the bluetooth thing is pretty curcial.. I have a JBL Clip, which is awesome, and I like having the speaker away from the device. It does have a cord, about 8" or so, which I can use without issue through the headphone output of the phone. But when I try to play songs with it connected to bluetooth it stutters and freezes, ultimately telling me that it can't be played

I know it's not a system issue because the phone connects right up, and there are no issues using the Clip with either the phone, or the Groove player.
So any ideas on how to fix this, or if a solution is being worked on for an update?