Good discussion question. I'll have to go back and look closer at the video, but I actually do what you do- clutch in, down a gear, clutch out, clutch in, down a gear, clutch out, etc. My gear indicator may not catch up until I'm done down shifting but you should hear the multiple down shifts. I also rely on the slipper and don't blip the throttle to help speed up the process.
I'd be interested in others sequence and approach to these types of corners too (including braking) as it's been something I have been working on this season.