I have quite a bit of census information and will and other legal information pertaining to Tate. His father was named Joseph, and he almost certainly had a brother named Joseph. Tate and most of his brothers and sisters were much older than your Joseph, and the 1810 census doesn't show Tate's father Joseph with any males under 10. Tate is in my wife's family, not my family.
Given that some of what I plan to send is conjecture, would you prefer to make the exchange by email so as not to spread possibly false information? If your Joseph is the one who married Elizabeth Peel, you have my email address. (If you have lost it and it doesn't display here, find me on the Alexander DNA site.)