No, they do not claim copyright over the physical bust.
The article said they actively blocked others from even taking photos.
"The museum has long prohibited visitors from taking any kinds of photographs of its biggest attraction."
So they are asserting copyright of the work in the sense they are restricting access.
Their only real reasonable excuse might be that flash photography is likely to damage the work but non-flash photography should still be allowed. They've got history of blocking all such access.
Also, I wonder if the "flash" on a phone is comparable to older types.