Yes, there is a Christian theological model for a non-literal ark. Such a model is provided and developed in great detail by Emanuel Swedenborg (1688-1772) in his massive work Secrets of Heaven, originally published in eight Latin volumes in London, 1749-1756.
In Secrets of Heaven #554-1059 Swedenborg offers a verse-by-verse spiritual exegesis of the Flood story in Genesis 6:1-9:17.
The symbolic nature of the early chapters of Genesis
Swedenborg interprets the characters and events of the first ten or eleven chapters of Genesis as referring, not to literal events of actual historical human beings, but rather as symbolic or "correspondential" accounts of whole races and generations of early humans. In his interpretation, Adam and Eve, for example, do not refer to two human individuals named "Adam" and "Eve," but rather to an early culture of human beings. They represent the first beings on earth who were both physically human and had a developed spiritual awareness of God and heaven.
The undivided mind of the first truly human culture on earth
These early humans, according to Swedenborg, had an undivided mind, such that whatever they wanted in their hearts, their minds and their hands immediately followed. They were in a state similar to human infants, who have no ability to filter their thoughts and desires, but every thought and desire flows immediately into outward expression.
This state was both their beauty and their downfall when their desires began to become selfish, worldly, and evil rather than good, innocent, and focused on God and spirit.
The spiritual meaning of the Fall of Humankind
Swedenborg interprets the Fall of Humankind in Genesis 3, not as a literal eating from a forbidden tree, but rather as these early humans' decision to follow their own ideas based on sensory information and pleasure rather than following the voice of God teaching and guiding them from within.
The decline of humanity recounted in Genesis 4:1-6:8 was especially rapid and disastrous precisely because these early humans had no ability to filter their thoughts and desires, but immediately expressed everything they felt, whether good or evil. So when their desires became evil, they rapidly became utterly corrupt from the inside out.
This set the stage for the spiritual events narrated symbolically in the story of the Great Flood. Swedenborg summarizes the spiritual state of this corrupted early human culture in this way:
In specific regard to the people of the church before the Flood, they conceived appalling delusions as time passed. The goodness and truth that belong to faith they merged so thoroughly with their foul desires that almost no trace of either was left to them. When they reached this point, they virtually suffocated themselves. A person lacking any remnant [of goodness or truth], after all, cannot survive. (Secrets of Heaven #560)
The Flood, then, was not a physical event, but rather a spiritual event in which a flood of rampant, unchecked evil desires and false thoughts spiritually suffocated the people of these early corrupted human cultures—which Swedenborg believed led to their physical extinction as well.
Noah represents a new phase of the human mind and spirit
The story of Noah and the ark, in Swedenborg's interpretation, is the story of a remnant of that early race of humans that was not so corrupted, and that survived and continued forward through a fundamental change in the nature of the human mind and spirit.
That change, to put it in modern terms, was the ability to compartmentalize their desires and thoughts, so that they could want to do something, but through the exercise of a distinct and separate reasoning capacity, prevent themselves from acting on that desire, but do something else instead.
In the life of an individual human being, this change of spiritual state corresponds to the psychological change that takes place when we make the transition from our infant and toddler stage to the stage of early childhood, usually somewhere in the age range of 3-5 years old. Before that transition, we immediately express everything we feel and think. After that change, we are able to mask our true desires, and present to the world a different face, and engage in different actions, than the ones our heart desires in the moment. We can now dissemble and lie about our true thoughts and feelings. We can now also stop ourselves from acting upon desires that we know are wrong or will get us into trouble.
In Swedenborg's interpretation of the Flood story, this new divided mind is represented by the ark itself, with its three floors and its rooms. His exegesis of the rooms in the ark is too lengthy to quote here. This excerpt provides a taste of his interpretation:
Since the Lord foresaw, then, that if the human race continued in this tendency they would succumb to eternal ruin, he provided that the will should be split off from the intellect. . . . The people of this church had to reform. The side of a person called the intellect had to reform first, before it was possible for the other side, referred to as the will, to do the same. So the present passage tells how the contents of the will were separated from those of the intellect and how they were concealed and stored away, so to speak, in order to block off any stimulus to the will. (Secrets of Heaven #640-641)
In short, Swedenborg interprets the "rooms" in the ark (Genesis 6:14) as a separation of the human will, or heart, from the human intellect, or thinking mind. The purpose of this was to make it possible for fallen humans, whose hearts had been corrupted, to be reformed through the exercise of their thinking mind.
In plain terms, humans could now learn intellectually what is right and wrong, and impose order upon their corrupted heart by stopping themselves from acting upon their evil impulses, and obliging themselves instead to act according to what is good and right. This happens, of course, by learning the truth as taught by God in the Bible, and by acting from God's power rather than from our own power.
In Swedenborg's interpretation, the story of the Great Flood and the ark of Noah is the story of a major transition both in the early psychological and spiritual life of an individual human being (the transition from infancy to childhood) and in the early life of humanity as a whole.
Ever since that critical transition, we humans have not been in a state of spiritual oneness (or "flow," to use a modern psychological term), but in a state of separation of heart and mind. This state is necessary for our spiritual regeneration, or rebirth, to take place in fallen and corrupted ("sinful," to use the Biblical term) human beings.
For a fuller account of the deeper meaning of the story of Noah, the Flood, and the ark, addressed to a popular audience, please see my article, "Noah's Ark: A Sea Change in the Human Mind."