But what does it mean for a moral claim to be simply true, full stop?

Well, in my world, it means that the premises are built into saying "moral claim"; that the subject matter of "morality" is the implications of those premises, and that moral claims are true when they make true statements about these implications. If you wanted to talk about the implications of other premises, it wouldn't be the subject matter of what we name "morality". Most possible agents (e.g. under a complexity-based measure of mind design space) will not be interested in this subject matter - they won't care about what is just, fair, freedom-promoting, life-preserving, right, etc.

This doesn't contradict what you say, but it's a reason why someone who believes exactly everything you do might call themselves a moral realist.

In my view, people who look at this state of affairs and say "There is no morality" are advocating that the subject matter of morality is a sort of extradimensional ontologically basic agent-compelling-ness, and that, having discovered this hypothesized transcendental stuff to be nonexistent, we have discovered that there is no morality. In contrast, since this transcendental stuff is not only nonexistent but also poorly specified and self-contradictory, I think it was a huge mistake to claim that it was the subject matter of morality in the first place, that we were talking about some mysterious ineffable confused stuff when we were asking what is right. Instead I take the subject matter of morality to be what is fair, just, freedom-promoting, life-preserving, happiness-creating, etcetera (and what that starting set of values would become in the limit of better knowledge and better reflection). So moral claims can be true, and it all adds up to normality in a rather mundane way... which is probably just what we ought to expect to see when we're done.