Only that person can tell you the pronouciation (s)he prefers .
There was an Irish poet named William Butler Yeats (without the last "e") that is known to rhyme with "gates" but that may be precisely why your Yeates has retained that extra "e" - to encourage a pronounciation to rhyme with "beets". But I would ask the person in question.
I write this as I am watching reruns of Keeping Up Appearances on TV. The central character's name is Mrs Bucket, but she insists that everyone pronounce it Boo-kay (a French affectation).
The double pun on this is that she and her sisters are all named after flowers: Hyacinth, Rose, Violet ... Together they make a "bouquet".