There are primarily three non-theistic routes to try to defend moral realism.
First, there's Mill. In your question, you seem to allude utilitarianism in stating "Believing in objective moral values might be a good strategy to maximize happiness" but then you kind of depart in suggesting that this would not make them true.
For Mill, the word good means pleasure-causing, and the word bad means painful. If there are objective things that would increase or decrease pain or pleasure, then it seems there's an objective account of good and bad along this definitions. This could be called "moral realism." Admittedly, there's a grounding objection ready at hand which is why would what makes these ideas (pleasure and pain) equivalent to ideas we call "good" and "bad." (I will return to this problem shortly). But for Mill, the only keep assumption is that we should relate pain/pleasure to good/bad.
Second, there's Aristotle. I read him as a type of moral realist. On his configuration, what is good is that which accords with the kind that something is. I.e., what makes a dog an excellent dog. This becomes ethical "good" when used in reference to what makes a human an excellent human and enables their flourishing. If there are any facts about human flourishing, there's a reason to view these things as real and thus what this theories declares good or bad to be real. Again, this view might leave us wondering what is different between saying "a good human being" and "a well-functioning human being." In which case, it seems like we could drop the predicate. For a contemporary reference look at "Modern Moral Theories" by G.E.M. Anscombe.
Aristotle actually has a nice objection to crude readings of Mill (anachronistically) -- which is that some people have thoroughly wrong understandings of pleasure and pain due to warped upbringings, etc.
Third, there's Kant and similar theories. Here, "good" means in accordance with objective reason and bad means done subjectively rather than for objective reasons. (I'm sketching quickly here because the details are not relevant to your question).
To put it another way, your question involves a semantic problem as well as a metaphysical one. On the semantic level, it matters greatly what we mean by "good" to answer whether or not, such considerations of the good can actually exist. It seems pretty obvious that there are at least some aspects of pain/pleasure that do exist in the world and actions can occur relative to these. Similarly, it seems pretty obvious that there are animals with certain arrangements that are better or worse for them -- e.g., just ask the cats of vegans.
What often seems to be at work in this sort of question is the rejection of a separate metaphysical category of moral properties that are real. And this would be a deep objection to Kant where such properties do seem rampant.