It's one stop darker, but you can save some money and get the 70-200mm f4. I use it very comfortably with my D600. That said, I've used the 28-300 with a D600 shooting concerts and it's a usable combination.
Here are a couple non-equipment related things. You're shooting horses indoors at a distance, and you want to shoot in bursts, so that's going to demand setting the camera so that it either works for you, or just gives up and shoots dark. I recommend setting up your camera as follows.
- Turn the lens VR on
- Set the camera to shutter priority and set the shutter speed to 1/125sec, IMO the slowest reasonable speed for trying to capture what you're trying to capture.
- Turn on Auto ISO and set Max ISO = 6400 (other settings won't matter in Shutter Priority)
Fire away. The camera will probably always set your aperture to wide open, which may cost you a little sharpness, but you'll stop the action. Check you pics for both light and motion. If they are too dark then step down your shutter speed to 1/60 and try again. If that doesn't work step down to 1/30, etc. At some point you'll either find a usable shutter speed or realize that it's just too dark to shoot with what you have.
If you hit this point (i.e. blurry when exposure is right, dark when motion is stopped) then you need to determine if this is "fixable" with a new lens. Take note of the shutter speeds that fix each problem - the max speed where exposure is right and the min speed where motion stops. You're zoom is a f/5.6 at the long end, so let's assume that's the aperture the camera used. The 70-200mm f/4 will be one stop brighter, and the f/2.8 will be 2 stops brighter. What that means is that if the difference between the two shutter speeds you noted is 1 click then the f/4 will work for you, if it's 2 clicks then the f/2.8 is what you need. If it's 3 or more clicks then you're in an unworkable situation (i.e. you can't get glass fast enough to shoot that fast in that light) and the best you can hope for is to shoot RAW and fix them in post as best you can. Fast glass is nice, but don't go spending money on lenses to fix a problem you can't really fix.