I am having the same problem.
The official English name of my tribe is the Crow Tribe of Indians and, as suggested, I am, for the time being, using the name of the tribe-Crow-as the last name. However, it is of no use whatsoever when doing a search. First of all, there are so many Crows to sift through and so far none have been connected with the tribe? Secondly, I have found no documents for any of my ancestors with the last name of Crow. If I search using their Indian name such as Sarah Strikes the Gun, I have better luck. I wonder also if I should have used the Indian name of the tribe which would have been Apsáalooke instead of the English name.
I plan to go to the Crow Indian Fair this summer so, hopefully, I will have time to talk to someone on the tribal council to see what they would prefer I do.
Hope you reach a workable solution.