I'm from the John Nathaniel branch. I've only been able to locate that Major's parents were born in North Carolina. It took me a long time to verify that "Major" was actually his given name. I was able to locate a picture of his gravesite, so I feel pretty certain it is.
I know there was a previous post, from quite a while ago, that was also looking for information on Major. I don't think it ever got any responses.
I have not heard of the Bennett Museum, but I'll go take a look and see if I can find anything.