The Festival of the Lion King Live Stage Show in the Harambe Theatre at Disney’s Animal Kingdom does allow flash photography.
I tried a different approach on my last viewing of the show. I put my camera in Manual mode and set the shutter speed to 1/125th of a second, the aperture on the Nikon 28-300VR lens I set to wide open at any focal length and changed to Spot Metering. Spot metering allows the camera to get exposure information from a small part of the frame. My camera is set to 11%. I let the camera calculate the needed ISO for a good exposure.
Even at 1/125s, fast movements by the performers will still cause blurring. I wait until there is little to no movement. It helps to be familiar with the show as I have seen Festival of the Lion King many times. One can use youTube to watch videos of shows you may not be familiar with. That was how I was able to get this photo of one of the performers singing. He had just completed a movement and I knew he would be close to stationary for a few seconds.

Singer performing during the Festival of the Lion King live stage show.
Nikon D750/28-300VR, 1/125s, f/5.6, ISO 5000, 0 EV, 300mm Focal Length, Cropped.
Notice the very high ISO number of 5000. In processing this photo, I used Noise Reduction software to eliminate the digital noise which was apparent in the original image. I point this out as the next photo was taken at ISO 2200 which almost does not need any noise reduction for an image shared on the Internet. If I was to print it, I would definitely apply some.

Singer performing during the Festival of the Lion King live stage show.
Nikon D750/28-300VR, 1/125s, f/5.6, ISO 2200, 0 EV, 300mm Focal Length.
With today’s digital cameras, high ISO photography is not the burden it was just 5 years ago. I might try even faster shutter speeds in the future to see how much I can push my equipment. Have you tried this at Disney or at any other kind of performances?
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