Is there any Christian theological model for a non-literal ark?

The answer to your question largely depends on how you define "non-literal". Do you use non-literal in the sense of meaning "did not actually exist or happen?" If so, then yes. There is a model for this - the Framework view of Genesis. The referenced link largely deals with Adam and Eve, but the concept is easily extensible to the flood narrative and would largely center on the Epic of Gilgamesh and the Epic of Atra-hasis were a similar comparison of flood narratives undertaken. This view is the belief that the first parts of Genesis present a literary framework for a corrective polemic against a background of a rich body of Mesopotamian myths and should be read in that context - not as a journalistic account of a worldwide flood. In that sense, the ark may not literally have existed. Genesis can simply be read for the obvious theological meanings and the question of weather it was a real event simply doesn't matter.

On the other hand, this view advocates for a literal reading of genesis in which the author of Genesis intended for it to be interpreted as a literal flood with a literal Ark and not as metaphor. In this sense, it does not present a model for a non-literal reading of thestory of Noah's ark.

My understanding is that this view has been around for many years, but did not become popular until the publishing of The G3N3S1S Debate by J. Ligon Duncan, David W. Hall and Lee Irons in 2001.

Biologos ("leading the way in setting the tone for thoughtful and productive dialogue on the topic of harmony between science and faith") provides a theological model in 'How should we interpret the Genesis flood account'. It says that as early as the first half of the 19th century, geologists and theologians Edward Hitchc**k, Hugh Miller and the Rev. John Pye Smith viewed the emerging scientific evidence not as a threat to faith, but as an occasion to reach a better understanding of Genesis.

Biologos says the language used in Genesis 6-9 does not insist that the flood was global. First of all, the Hebrew kol erets, meaning whole Earth, can also be translated whole land in reference to local, not global, geography. Moreover, in this period of history, people understood the whole Earth as a smaller geographical area. There is no evidence to suggest that people of this time had explored the far reaches of the globe or had any understanding of its scope. Assuming that the Flood was local, its location has not yet been precisely determined. Though excavation of flood deposits in Mesopotamia provides evidence of ancient flooding, there is no evidence that it is unambiguously the biblical flood.

Biologos says that in the 1990s Columbia University geologists William Ryan and Walter Pitman concluded that a massive local flood took place in the area we now know as the Black Sea. They theorised that when the Ice Age ended and glaciers melted, a wall of seawater surged from the Mediterranean into the Black Sea. This flood, which may have occurred around 5500 B.C., would fit into the Old Testament timeline of Noah’s Flood. Robert Ballard, famous for finding the Titanic, led a 1999 expedition with the hope of finding more evidence for this theory. The expedition revealed an ancient shoreline for the Black Sea, and after radiocarbon dating, the findings supported their hypothesis that a freshwater lake and surrounding manmade structures were in place before the flood. Conflicts with the Black Sea explanation do exist, however. For example, 5500 B.C. is too early for Noah to have used metal tools to create the ark, and the location of the Black Sea does not fit the Sumerian and Babylonian accounts of the flood, which strongly suggest that it took place in Mesopotamia.

Ian Wilson supports the Black Sea hypothesis in Before the Flood: The Biblical Flood as a Real Event and How It Changed the Course of Civilization, covering detailed archaeological, historical and literary research. He resolves the Mesopotamian anomaly by explaining that the Black Sea inundation necessitated migration from the flooded region, with some of the survivors reaching the Near East. Wilson says that those who escaped this catastrophe would have fled on foot, not by boat. On page 134 he describes the impact it would have had on people:

Those living closest to the Bosporus would have become immediately affected by, and perhaps swept away by, the devastating in-rush of water. For those further along the shores towards to the east the first sign was most likely a distant roar, accompanied by an ominous vibration. The roar would have become louder and more incessant, as unusual amounts of flotsam began to appear in the lake-water – probably including whole trees and animal, human carcasses. As the lake level rose inexorably, concern must quickly have escalated to alarm. With a lake-level rise that Ryan and Pitman estimate to have been at least 15 centimetres a day, whole settlements would all too swiftly have become inundated, with even the most orderly evacuation turned into rout.

We now have a plausible explanation, consistent with the biblical account, of a local flood, one potentially supported by science, but with no opportunity to construct an ark before being overwhelmed by the floodwaters.

Even in the context of the Black Sea inundation, it is a hypothetical possibility that a survivor named Noah escaped on a boat (or ark) that must have already existed. For this to be a plausible case, the name of Noah should be linked in some way to the very earliest legends of the flood. Another famous story of the flood is in the Epic of Gilgamesh, which is both older than and remarkably similar to the story of Noah's Ark. Many scholars believe that the story of Noah's Ark came out of Gilgamesh, while others believe that they both evolved from a common, even earlier legend. Either way, Noah and his ark disappear as we trace the Near Eastern flood legends back in time.

I don't know of a detailed exposition that focuses specifically on the story of the ark, but in the Medieval ages it was common to say that each piece of scripture possessed the following levels of meaning:

• Literal sense
• Spiritual sense
• Fuller sense

From the link above regarding the "literal sense":

The literal sense is not to be confused with the "literalist" sense to which fundamentalists are attached. It is not sufficient to translate a text word for word in order to obtain its literal sense. One must understand the text according to the literary conventions of the time. When a text is metaphorical, its literal sense is not that which flows immediately from a word-to-word translation (e.g. "Let your loins be girt": Lk. 12:35), but that which corresponds to the metaphorical use of these terms ("Be ready for action"). When it is a question of a story, the literal sense does not necessarily imply belief that the facts recounted actually took place, for a story need not belong to the genre of history but be instead a work of imaginative fiction.

When one looks at the genre of the flood story, it (arguably, anyway) does not follow many patterns for a historical story - for example, it does not make clear when the event happened in relation to other well-known events as many history-genre stories do, ("In the Year X of the reign of Y", often referring to multiple rulers to make it clear), and it does not cite witnesses that can be cross-referenced (as, eg, the Gospels do).

The above link goes on to say

One of the possible aspects of the spiritual sense is the typological. This is usually said to belong not to Scripture itself but to the realities expressed by Scripture: Adam as the figure of Christ (cf Rom. 5: 14), the flood as the figure of baptism (1 Pt. 3:20-21), etc.

So, in this way of interpreting the Bible, the flood story is one example of something that prefigures baptism, and one example of something from the Hebrew Scripture pre-figuring something in the new.

It is an interesting example in terms of apologetics because this way of interpreting the Bible was formalized in the Medieval period (and can be seen as early as Origin of Alexandria), and shows then that non-literal ways of reading the Bible predate the emergence of modern science and the scientific method at least by centuries, and therefore refutes many naive ideas non-Christians have about Christianity.