If weight is major concern, you might also consider the Tamron 24-70mm 2.8. It has excellent sharpness, and with the crop factor it could replace several lenses in your bag. I even knew a guy who very successfully filmed a documentary with it. He swore that he wouldn't mind if someone welded the lens to his camera, as he's never taken it off since purchasing it.

As for your D7200, I'm not sure if you were referring to the shooting or autofocusing speed, as the lighting wouldn't affect the frames per second. If you go into our camera's settings menu, under "Autofocus" you should see two separate entries labelled AF-C/AF-S Priority Selection (probably under a1. & a2. respectively). Set these both to "Release" instead of "Focus". This will ensure that the shutter will continue to fire even if the camera doesn't believe it has attained focus.

You're probably already aware, but a faster memory card can also help with stuttering issues. Another trick you might try is changing the RAW bit depth to 12-bit lossless compressed. This will help ensure that the frame buffer doesn't fill up so fast, and will allow you to shoot continuously for longer.

I'm not sure how much of this you are already familiar with, but let me know if you have any questions.