According to one objection to arguments from evil, the existence of evil presupposes the existence of God, since objective evil could not exist unless God exists. My advice to critics of arguments from evil (AE) is this. Don’t use this objection unless you plan to acknowledge and address the obvious rebuttal.
AE may be understood as a challenge to the internal coherence of a theistic worldview. An AE can be understood as saying something like the following:
Look. You theists believe that X, Y, and Z are evil. You theists believe that God is good. You theists believe that good persons are opposed to evil. So you theists need to explain why a god who is good (in your sense of ‘good’) would allow so much apparently pointless evil (in your sense of ‘evil’). If you can’t explain it, then that is a problem for the internal coherence of your worldview.
When AE is understood in this way, it doesn’t presuppose that there are objective moral values.