I also found that quite a few were integrated into white society, becoming, "invisible", in the records as Native Americans. It seems there was a US government assimilation project designed to "civilize" Native Americans into European culture, (why didn't I hear about this in high school?), from the time of George Washington!
I learned too, there is an excellent chance I will never find the information I seek. Native Americans were not enumerated in the census until New York did one of their reservations in 1845, you can find it here. Of course I searched it, but found no Carters. I did find that many Native Americans went by very European sounding names, one census even gave their original name along with the Americanized one, others gave only their Native name.
Adding to the difficulty, there was a racial component. In the 19th century, some people felt embarrassment at having a Native American in the family tree and tended to hush it up. George on the other hand embraced his ancestry, in fact his obituary says his father Silvester was born on a reservation, and that George himself went to live on one for a time.
Which presents yet another problem, there were no reservations nearby. The Senecas left Seneca Point, moving to western New York, around 1779 when General Sullivan and his troops marched through destroying their villages and crops. This was 38 years before George's obit says his father was supposedly born there in 1817. Perhaps there was a small settlement of Senecas who had drifted back to their ancestral home, or never left the area and that was where Silvester was born and George went to live?