The exterior cladding is the least of that museum's concerns. The interior is where most of the problem lies.
As has been said many times before in this thread, museums and a lot of glass do not mix well. Unless everything on display inside is made of stone, the effects of light are to fade everything.
Why they have the fabrics/costume galleries up on the top floor of the crystal I will never know. That room has amazing windows, but they are now entirely blocked. It's completely idiotic. They need to find a new spot for that gallery, and then move something like mineralogy up there that won't be affected by light, and open up those windows.
So spandrel then.