It doesn't beg the question, "begging the question" is a type of fallacy in which we insist upon presuming X and then rely on that to prove X which is a waste of everybody's time. But yes, it is reasonable if we notice that Atwood's "The Handmaid's Tale" is Science Fiction to expect that awarding a Booker to its sequel "The Testaments" would constitute a Booker for Science Fiction.

I would imagine the distinction attempted here is that despite writing those novels, Atwood doesn't consider herself to be an SF author, perhaps Harrison says he is an SF author. Atwood also won for "The Blind Assassin" which very much is not Science Fiction, it's historical fiction, it's not history but all the stuff happening in the background was real.

This shades over somewhere. If your story features an England ruled by literal Nazis it's Science Fiction (alternate history is a type of SF, "What if the Nazis won?" is a fairly popular theme) but if your England just has some Nazi sympathizers and your story is set in the 1930s it might just be historical fiction.