if he was alive 1/1/1937, he would have filed a social security application and submitted a delayed birth certificate to show proof of age.
you have to ask for the delayed birth certificate because, as you found out, they would have to look in a different time period to find it.
if he was an enrolled member of the tribe, there might be something in his file to indicate birth, parents, etc. i have seen documents like this.
you should get a copy of his death certificate, so you might have more information. an obituary might help. you can see your local public library for that – they have the interlibrary loan program. be sure you know which state might have the delayed birth certificate.
suzanne hamlet shatto