In fact they're probably too generous with the "Sir", because he made his will as John Dudley esquire, and he's called the same in his IPM.
https://archive.org/stream/completepeerageo04coka#page/480/ (page 480 footnote a)
https://discovery.nationalarchives.gov.uk/details/r/C10241400
He wasn't a Baron - his father was the 1st Baron, and his nephew Edward was the 2nd.
Some of the Suttons did take to calling themselves Dudley, and this branch seem to have used that name exclusively.
It didn't do them any good - when Elizabeth's grandson became Duke of Northumberland, the story was spread that he was a pleb descended from a carpenter.