the original owner was able to
  own the piece in the first
  instance on very shaky grounds

The Migratory Bird Treaty Act [1] and Bald and Golden Eagle Protection Act [2] both allow permits, issued by the Fish and Wildlife Service, for possession of the protected birds.

This is used, for example, to allow museum exhibitions and Native American religious ceremonies. There's even a special form [3] to apply for the permit!

So long as the eagle is in a museum that meets the requirements of the eagle exhibition permit, the ground seems pretty firm to me!

[1] https://en.wikipedia.org/wiki/Migratory_Bird_Treaty_Act_of_1... [2] https://en.wikipedia.org/wiki/Bald_and_Golden_Eagle_Protecti... [3] https://www.fws.gov/pacific/eagle/permit_types/exhibition_de...