I don't remember Harris ever even trying to bridge the is/ought gap.

He addresses the argument quite explicitly. His point is that science always begins from some presupposed, philosophical value; physicalism and empiricism for physics, health for medicine et cetera. Harris's philosophical case rests on the argument that for any ethical 'theory' to make much sense it must be founded on the well-being of conscious beings; as in, nobody could sanely argue for a moral system which reliably brought about pain and suffering on a wide scale. Harris elucidates this point with a thought experiment: imagine a button which one could press which would bring about the worst possible suffering for everybody. If words like 'evil' are to make sense, they must refer to the sort of person who would push that button.

It's like Hume Fork, or the separation of 'relations of ideas' from 'matters of fact'. If Hume's guillotine is correct, then Hume's Fork falls because it is necessarily a presupposition laden with values about how we ought to conduct our search for knowledge. Hume essentially contradicted himself.

Harris's ethical foundation is closer to Aristotelian eudaemonia, rather than Bentham's utilitarianism. His argument is that, essentially: the well-being of conscious beings is the only metric for ethical action which makes sense -> facts about the well-being of conscious beings are non-arbitrary and can be discovered -> the empirical study of these facts can tell us how we ought to act in a given situation.