Maybe I can share what might happen in Oklahoma. In Oklahoma, having a tribal membership card will not help you at all - you must have a CDIB card (Certificate of Degree of Indian Blood), which shows that you are a member of a federally recognized tribe. To the best of my knowledge, none of the Cherokee organizations in Missouri or Arkansas have ever achieved federal recognition. Which I know is a real drag - my direct ancestors came from eastern Tennessee on the Trail of Tears but they stopped in Missouri when the eldest male died - and, because they did not make it all the way to Oklahoma until much later, I am part Cherokee by heritage and can prove it but I do not have a CDIB and never will - if your lineal ancestors did not reside on Cherokee lands when the Dawes Rolls closed around 1900, you're out-of-luck. The tribes maintain that your ancestors lost citizenship when they left tribal lands.
There are indeed blood quantam requirements for many Native American scholarships. Things are different in the Native American world - even if you were adopted by a Native American family just after birth, you are not considered to be Native American unless you can prove blood. The NA scholarships I've seen all require 1/4 quantam. There is a reason for this - I have actually seen people with quantams less than 1/1000 trying to claim tribal benefits.
So, you could apply as a Native American URM and, in many places, your self-reporting might not be questioned - but here in Oklahoma you'd be asked for your CDIB first thing.
