I don't know if you're Christian or if the Christian worldview is of interest to you, so apologies if that's not the case but I'll try to give a Christian view.
First of all, what aspect of suffering are you saying is incompatible with God's existence? Physical pain, for example, isn't something I would classify as a moral evil. It's necessary for survival. Babies who are born with a condition that causes them to be unable to feel pain usually don't survive beyond a few years because they can do terrible damage to themselves without knowing it. The same is true for animals.
According to Christian belief, moral evil is not part of God's original plan but is a reality of living in a fallen world in which we have free will, and frequently use it to cause suffering to ourselves and other people.
I'm not sure you should be working with the assumption that an easy, low-stakes and pain-free existence is necessarily what God wants for us anyway. There's not much opportunity for courage, compassion or even joy in that version of life IMO. But staying with physical suffering, illness and death - even the physical world is affected by the Fall, bringing a different level of pain and struggle and ultimately death into the world.
In Christianity Jesus is the central 'answer' to the issue of suffering because far from being remote and uncaring, God experienced all of our suffering with us and ultimately triumphed over both death and sin with all its attendant suffering. So while we're still living with that reality, we look forward in faith and hope for a time when every tear will be wiped away.
I don't know if this is helpful, I know it's a very emotional issue and ultimately these answers are rather dry. I think there are philosophical ways in which we can understand things more clearly, but our response to people who are actually suffering should always be to try and relieve that suffering and to love and care for them.