Re God deciding to establish a system of morality: Presumably this amounts to God making a bunch of moral claims, and then us defining “morality” as “the system of moral claims made by God”. But I don’t see why this system should have any definitive relationship with what we ordinarily talk about when we talk about “morality”. After all, our usual talk of morality is intricately connected with how we plan what we are going to do—we hopefully plan to do moral things. Now maybe we also plan to follow God’s system of morality, either because we’ve decided to submit to his omnipotence or because we think his system of morality is a rather good one. But these are not necessary relations but rather practical or moral considerations. In other words, we follow God’s will because God is moral (or we don’t follow his will because he isn’t moral), rather than morality being defined by God’s will. This distinction is important to make in case we want to make sense of moral criticism of God for Old Testament atrocities or other possible moral failings of his, even if we believe in God.
So no, I don’t think that God’s existence would bear any relation on whether morality is objective.
Re God deciding to establish a system of morality: Presumably this amounts to God making a bunch of moral claims, and then us defining “morality” as “the system of moral claims made by God”. But I don’t see why this system should have any definitive relationship with what we ordinarily talk about when we talk about “morality”. After all, our usual talk of morality is intricately connected with how we plan what we are going to do—we hopefully plan to do moral things. Now maybe we also plan to follow God’s system of morality, either because we’ve decided to submit to his omnipotence or because we think his system of morality is a rather good one. But these are not necessary relations but rather practical or moral considerations. In other words, we follow God’s will because God is moral (or we don’t follow his will because he isn’t moral), rather than morality being defined by God’s will. This distinction is important to make in case we want to make sense of moral criticism of God for Old Testament atrocities or other possible moral failings of his, even if we believe in God.
So no, I don’t think that God’s existence would bear any relation on whether morality is objective.