My opinion would be to keep using "Thatch" as the LNAB and keep using "Teach" as the "Other Last Name(s)".
I only state this because it seems (to me) that, even though both last names are referenced, it appears that "Thatch" is more prevalent than "Teach" and others.
The following is a page (via an Ancestry sharable link) from an Oxford University Press in London, England.
The book is called "The Dictionary of National Biography" and, more specifically, it's from "Volume: 22 - Page: 1281" of that book.
It may not be a ton of information, but it does look like it's just as valuable.
Just a little helpful input/option that I hope helps you out in some way.
Good Luck! :-)