Cyril of Alexandria argues that the Word of God the Father suffered in his own flesh, precisely because "in his own flesh" is where the Word can suffer (On the Unity of Christ). The flesh is where the pain receptors are, where nerves and neurons live, and where damage can be done. By nature, God is Spirit and his Spiritual Being transcends heaven and highest heaven (John 4:24; 1 Kgs 8:27).
If someone wants to say that God indeed suffers not in the flesh, then ask "how"? And in what way? I am not sure how to affirm that, although I agree with Origen who says that God somehow suffers love in his Commentary on Ezekiel.
But note that Origen speaks of compassion (suffering together with) in this way: he says like a doctor whose compassion leads to the healing of the patient, so God's suffering is a compassion that touches to heal; we don't give God our illness when he heals us, but we receive his health (see his On First Principles).
This paradigm approximates how Jesus healed. Jesus did not get leprosy, but he healed it. That said, Jesus also had flesh, and so he hungered, thirsted, and experienced all sorts of suffering. But that is precisely because he is the Word of God the Father who suffered in the flesh for us and for our salvation.