I’m not so sure soundproofing would help in this case. His video appears to be taken approximately 15 feet from the landing doors and that snap is loud. I suspect that the box shape of the lobby plus the cabin being at any floor above 5 is going to make it all the worse as the sound may be amplified for him the higher up the cab is.
But I agree it might be a fruitless effort to get the brake to pick quieter.
I had 5 of these on my route. KONE MRL’s. And that same brake pick/drop sound was just “what it is” but the lobby’s were outside not on the interior of the building, but you could hear it in the cab and at each landing and at every run.

I’m curious to see if the OP keeps us informed as to any headway towards a resolution.