The Arapaho nation is just that..a nation. They are (sort of) a sovereign nation (along with the Shoshone) within the US territory.
Although they have the right to self-govern, Trust Doctrine, which governs US-Tribal relations may have a bering on the issue of Eagles. The US has feduciary responsibility when it comes to trust lands (the US gvrmnt holds the deed of the land in trust) and management responsibilities in regards to resources associated with trust lands, including mineral, oil, water, timber and gas.
I don't know if animals are considered a "resource."http://www.namsinc.org/pdf/Soverign%20nation%20white%20paper%202.pdf
Tribal sovereignty in the United States refers to the inherent authority of indigenous tribes to govern themselves within the borders of the United States of America. The federal government recognizes tribal nations as "domestic dependent nations" and has established a number of laws attempting to clarify the relationship between the federal, state, and tribal governments. The Constitution and later federal laws grant local sovereignty to tribal nations, yet do not grant full sovereignty equivalent to foreign nations, hence the term "domestic dependent nations".
Trust Relationship. The federal government has a "duty to protect" the tribes, implying (courts have found) the necessary legislative and executive authorities to effect that duty.
Tribal sovereignty is dependent on, and subordinate to, only the federal government, not states, under Washington v. Confederated Tribes of Colville Indian Reservation, (1980). Tribes are sovereign over tribal members and tribal land, under United States v. Mazurie (1975).http://en.wikipedia.org/wiki/Tribal_sovereignty_in_the_United_States
I suspect the feds can stop the killing of eagles, if they wanted to, regardless of religious practices involved. The issue of religious practices has been central to Indian relations. Calling them to task on this will stir up a lot of anger.
Federal Indian Law
When we enter into the realm of "federal Indian law," we need to keep in mind that we are traveling in a semantic world created by one group to rule another. The terminology of law is a powerful naming process. In working with this law, we will use the names that it uses, but we will always want to keep in mind that the reality behind the names is what we are struggling over.
According to the theory of sovereignty in federal Indian law, "tribal" peoples have a lesser form of "sovereignty," which is not really sovereignty at all, but dependence. In the words of Chief Justice John Marshall in Cherokee Nation v. Georgia (1831), American Indian societies, though they are "nations" in the general sense of the word, are not fully sovereign, but are "domestic, dependent nations." The shell game of American Indian sovereignty -- the "now you see it, now you don't" quality -- started right at the beginning of federal Indian law. The foundation of federal Indian law is the assertion by the United States of a special kind of non-sovereign sovereignty.