What are the major problems that Chalcedonians have with Miaphysite Christology?

One nature or two? A Divine nature, in one person; or, Divine-plus-Human natures in one person? Which is it? A hypostatic union, or a miaphysite unity? Which is it?

Each perspective has something going for it. On the miaphysite side, God the Son has been one with the Father and Spirit for eternity. How could he be anything but God? Yes, Jesus became a human being, but being conceived by the Holy Spirit and born of a virgin, he was not a child of Adam and was therefore without sin. He was, however, the last Adam. In the first Adam, all die. In the last Adam, all who believe receive eternal life through a life-giving spirit, who is Jesus.

Here, I believe, is the primary weakness of the miaphysite approach to understanding who Jesus is: Jesus identified himself as being both the Son of God and the Son of Man. Jesus did not exist in eternity "past" as a flesh-and-blood human being. In the fullness of time, however, he burst on the scene according to God's plan for the ages, born of a woman, born under the law, so that he could redeem fallen humanity, for whom the Law had become their undoing and their condemnation.

We know that Jesus did not become God when he was conceived by the Holy Spirit in the womb of the virgin Mary, but he did become a human being, given by the Holy Spirit a genetic code that worked itself out in the normal process of gestation and growth to the point of becoming a full-grown man.

In that self-emptying way of becoming human and experiencing all the limitations and temptations the rest of humanity experiences, yet without sin, Jesus became a faithful high priest who in his flesh offered himself up to his Father as a once and for all sacrifice for sin.

To be sure, we observe in the Gospel narratives flashes of deity, as Jesus was led and empowered by his Father to perform wonderful miracles, demonstrating his authority not only to heal, to raise from the dead, and to command the wind and the waves but also to forgive sins, which has always been a divine prerogative.

By the same token, however, Jesus succumbed to weariness, seeking rest in the restorative power of sleep and in sitting by a well in Samaria, "tired as he was from the journey." In other words, Jesus experienced what God had never experienced before. YHWH of the Old Covenant could "neither slumber nor sleep," but was eternally vigilant, attentive to the needs and prayers of his saints on earth.

In short, divinity, rightly understood, cannot grow "in wisdom and stature, and in favor with God and man." Jesus, however, experienced all those things as both the Son of Man and the Son of God.

Perhaps the word nature is a stumbling block to those of the Eastern Orthodox faith. The Bible does not attribute two natures to Jesus, one human and one divine. The divine and human "sides" or aspects of his personhood were complementary, each with its own contribution to the life of the God-Man. Jesus was one person, not two, but where his humanity began and his divinity ended, or where his humanity ended and his divinity began, is impossible to say. Along with the apostle Paul, we need to say sincerely and humbly,

And without controversy great is the mystery of godliness: God was manifest in the flesh, justified in the Spirit, seen of angels, preached unto the Gentiles, believed on in the world, received up into glory (1 Timothy 3:16 KJV).

My understanding is that Divine nature and human nature cannot 'merge' but are united in the Person of Christ. The union is in his Person, not in an attribute : 'nature'.

I assume that I would be referred to as a 'Chalcedonian'.

From my own point of view, therefore, as a Chalcedonian, the situation is not reconcilible. The concept of Divine nature and human nature being combined (merged) into one 'nature' is not logical and I can think of no scripture that would be useful for someone arguing that it is so.

I would call that the 'main problem' (see the OP) with attempting to bring the two parties together.