My great grandmother was a full blood Navajo. We also have Cherokee blood in the family. I used to live in Tahlequah, OK, which is the capitol of the Cherokee Nation. I lived in Missoula, Montana, the area around and in which has the majority of 6 different tribes. I can safely say this.
American Indians prefer to be called American Indians. Native American is a PC invention by whites and it is cloying and untrue. We are NOT priestly hippies in commune with nature. We're just folks, like you. No one asked US if we like Native American.
No one asked US if we like ANY of the national sports teams with Indian type names. We would like to be asked once in a while. The Seminoles in Florida said 'go with it' and did the Redmen in Tahlequah. Chief Illiniwek at the University of Illinois made any grown up Indian want to puke. "Redskin" is the N work for indians, so imagine how many Indians are big fans. It would be like the Chicago team being call the Little Black Sambos instead of the Cubs.
As for Cleveland. Get rid of the cartoon. It would insult ANY race to be represented that way. Indians, Braves, Blackhawks, go for it.
But mostly, who would it kill dead to ask? How about polling the enormous indian population of 40,000? Would it mean you are less white if you asked?