Pepe,
It will not be a function of the Native corporations, it will be a function of the Tribal government which is different and separate from Native corporations.
 In the Lower 48, in most states, tribes have been doing this for years because they have their own tribal courts, tribal social workers and tribal foster homes. 
All federal laws have to be followed. Parents will have a legal right to representation just as they now have in State court. 
Tribes do have, under federal law, the right to be sovereign, this goes back to the days of treaties. As for who will look out for the children and/parents if the tribe violates their civil rights under federal law, it will be the federal government unde the Department of Justice, Civil Rights.
The immunity of the tribe from law suits......seems to me a few months back the Alaska Supreme Court made a decision on this but it was not covered in the Dispatch but it was cover in other media. But, you are right, there is a potential issue.