Shutter Speed or frame rate we looking at when dealing with flickering issue?
The reason you get flicker in the video is that many conventional lights don't actually stay on all the time.  When running on AC power, the power actually "goes off" 100-120 times a second (depending on if you have 50hz or 60hz AC).  The way this oscillation impacts things differs by type of light from a slight dimming in incandescent to a color shift in some florescent lights to other odd behaviors in LEDs (depending on the DC conversion circuitry and how power is supplied to the LEDs).

When the light says it is flicker free up to speeds of 1/3000th of a second, it means that any flicker it has should balance out for frames as fast as 1/3000th of a second (meaning the flicker, if it has one, is faster than that so that it's guaranteed to give most of the exposure in any 1/3000th second window.)

As far as the shutter speed the camera uses, I'd assume it is whatever setting you choose.  At 1000fps shutters, the "shutter" is purely electronic and is just the interval between blanking and reading of the sensor.  As long as each frame has more than 1/3000th of a second to accumulate light, you should be ok.  If your camera gives shutter speeds in degrees, that means you'd want at least a 120 degree shutter at 1000fps.  If it measures it in fractions of a second, just make sure it is set for 1/3000 or higher.

You may need to adjust the exposure type for HFR shooting by putting the camera in HFR mode and then using the function menu to make it either manual or shutter priority exposure so that you can keep the frame rate fixed at a slow enough setting.