To be completely fair, the Euthyphro Dilemma would still exist even after we remove God from the picture. After all, is an action immoral because the majority/government condemns it, or does the majority/government condemn an action because it’s immoral? We often see this debate among moral objectivists and moral relativists.

Back to your original point, I would use an argument from design to establish an objective basis for morality. In our world, we believe that child abuse is wrong, but the only reason we think that way is because God designed mankind in such a manner where child abuse is harmful to our species. God could have easily designed a universe where actions we normally deem to be “child abuse” would be the most effective way of raising children. Likewise, homosexuality is wrong for us because it defeats the purpose of God designing the two sexes. Unlike animals who breed and then abandon their young to fend for themselves, God specifically designed us to embrace the virtues of love, empathy, and family. While God designed nature to be competitive, He designed mankind to be civilized; that's why actions such as rape and murder are wrong for humans.

We don't need to worry about God arbitrarily deciding one day that rape and murder are acceptable because such actions are destructive to His design for mankind. It would be like God suddenly issuing an instinctual directive to cats to begin breeding with dogs; a cat simply isn’t designed to breed with a dog. That being said, I believe God granted us free will because the universe would be absolutely meaningless if He had ensured everything went according to design. And of course we already know the price of free will is sin. It would then be accurate to view sin as a spiritual force with the power to twist or corrupt God’s designs by introducing perverse, even wicked desires into the human soul, just like a virus that hijacks a host cell by inserting its genetic information.

Unlike us, animals were never designed for free will, but given a set of instincts instead. Such instincts are God's commands to them. They cannot sin because God deliberately programmed them to be incapable of acting beyond His design.