Here's the meat and potatoes of the argument:
Premise 1: Morality is a rational enterprise.
No, morality is a social enterprise. It's about how humans should encounter and interact with other humans. It's a mix of evolved tendencies and acquired behaviors.
Premise 2: Moral Realism is true, meaning moral facts and duties exist.
No, there are no absolute statements and that includes moral statements. The nearest thing to moral 'facts' are the evolved and learnt behaviors I mentioned, and from the observer's point of view they're not imperatives; the manner in which they are or aren't imperatives for the subject will vary greatly.
Premise 3: The moral problems and disagreements among humans are too much for us to assume moral facts and duties are grounded in a human source of rationality.
As I said, they've evolved to suit us as gregarious primates who benefit greatly from cooperation.
Premise 4: Moral facts and duties are grounded in a necessary, rational source (from 1, 2,3).
Not even close.
Premise 5: This source is what we call God.
The term 'God' has no definition appropriate to a real entity, such that if we found a real candidate we could determine whether it was God or not. Without that, the expression 'God' is meaningless.
Conclusion: Therefore, God Exists.
Nope.