I don't know the answer to this but I wanted to applaud for an excellent question, I didn't know how much I wanted to know about Nestorian until I read this post.

Great question! As you seem to know; the 'Nestorian' Church was only called that by the "western" Church, as they identified themselves as the Church of the East or the Assyrian Church of the East. When I now said "western" Church, I need to specify for those who might equate it with the Catholic Church, because in the sixth century the western Church includes the Orthodox as well. Though, speaking in those terms, "Orthodox" and "Catholic", is sort of anachronistic at that time. In the sixth century there is still one church with certain communities in the East starting to withdraw from the ecumenical councils as they did not agree with the compromises made there. Though, in the sixth century, the Church of the East was not completely out yet. To understand how the western Christians viewed the 'Nestorian' Church in the sixth century, we have to first understand the name "Nestorius" and what the westerners understood by that, second we need to understand the divide that started at the Council in Chalcedon in 451.

Nestorius was the Archbishop of Constantinople who got condemned at the council of Epesus in 431 for speaking of a christological doctrine that was too much for the rest to swallow. What he proposed was that there were to Persons (hypostases) in Christ; the Divine and the human. This had nothing yet to do with the Eastern Church. How could it? Nestoruis was a man from Constantinopel who spoke greek, and the language of the Eastern Church was syriac. Nestorius would become a somewhat prominent figure in the Eastern Church. The important thing here is that Nestorius were not condemned by the Eastern Church, and that his name "Nestorius" was used in the western part of Christendom as a label of insult to attack opponents.

We now jump ahead to the council of Chalcedon (451). This is where the Church of the East will part from the Church of the West, but not of reasons that you'd think. Because, as Sebastian P. Brock has argued for brilliantly, it seems that the reason was bad communication. I said earlier that the Church of the East spoke Syriac, and in the west it was mostly communicated in Greek, especially in these councils which were held in the Byzantine territory. At the council of Chalcedon there were a discussion about faith. The council was discussing if there were two hypostases in Christ or just one. Three key terms were then crucial for the discussion: nature, hypostases and person.

Brock argues that these were difficult words to translate from greek to syriac. The greek word hypostasis were translated to the syriac word qnoma. The problem here is that the syriac word of qnoma has a much wider range of meanings and would in english translate closely to "individual manifestation". Qnoma has again traditionally been understood wrongly as "person". This has then finally led to the classical definition of Nestotianism as "the belief in two persons in Christ".

As you see, the Church of the East became an easy target for the rethoric of the west in this trublesome communication. The differenses were really not to big between the two, but the language barrier was the deal breaker. So when you ask: how was the 'Nestorian' Church viewed by the western Christians in the sixth century? It is tempting to answer: "wrongly". The more correct answer I think would be that they lost contact, but the image remained. These were the people who believe that Christ had two hypostasis: one divine and one human.

Sources: BROCK, Sebastian P. (1996) The "Nestorian" Church: A Labmentable Misnomer. In "Bulletin of The John Rylands Library", vol. 78, nr. 3, pp. 23 - 35. BAUM, Wilhelm & Winkler, Dietmar W. (2003) The Churches of the East: A Concise History. London. Pages: 204.

Edit: A lot of spelling. I was kind of in a hurry when I wrote this.