The Chickamauga did not "tuck tail and run", they left the nation to continue the fight.
Not true. Some of the Chickamauga followers abandoned their Cherokee citizenship and moved to the west, some as early as 1790. Thier descendants live among the members of the Cherokee Nation of Oklahoma and the United Keetoowah Band of Cherokee Indians. There are NO Chickamauga descendants who don't have ancestors on the Dawes Rolls. Any claim otherwise is false and I would challenge you or anyone making such a claim to prove you have Chickamauga or any Cherokee ancestry.
Most of our Chiefs (the bloody seven) either died a violent death, or were assasinated by New Echota.
If you are speaking of the chiefs of the Chickamauga towns, such as DraggingCanoe and The Bench, those men belong to US and no one else. Their history is part of the history of the Cherokee Nation and UKB. I don't know where you get the "our" in your statement, but if you cannot prove that you have Cherokee ancestry, nothing about the Cherokees belongs to you.
The first step to lay claim to the history of a people is to prove you are one of the people. So far, I don't see any proof that you are one of us and have any right or claim to OUR history.