I have the exact same problem since el Capitan.
Bluetooth audio working smoothly when only my JBL Pulse 2 is connected. When I also pair my Magic Keyboard the stuttering begins and can only be fixed by disconnecting the keyboard.
There are many "fixes" that attempt to solve the problem on the web. Some happen to work for some people but only for a short time period.
I have yet to find a definite fix for this problem.
I SHOULD NOT have to disconnect my keyboard to listen to non stuttering bluetooth audio.
If you can link me some solutions I maybe haven't tried yet I'd be happy.