What are our blind spots? Basically, I would say that whenever we make assumptions but are unaware of them.
The grand thing about the scientific approach, I think, is that in collaboration with philosophy, the underlying assumptions can be made explicit and questioned from time to time. In so far, it is reasonable to assume that with scholarly methods we can prod reality and reality answers us back. That kind of naturalism indeed must be the foundation of the academic enterprise. For, unless we assume that there is a natural world and we can find out how it works, there really is no point in discussing anything.
As far the the knowledge gaps are concerned, I am going with Cynthia Stokes Brown who said that they will be part of the natural world too. So, there indeed is no need to assume something supernatural, just to fill in the gaps. (See Origins VI 1 for Cynthia's article on The Meaning of Big History - philosophically speaking)
Moreover, this position has the advantage that we surely can all agree on that the natural world exists and we are able to investigate it, but it does not at all require us (big historians) to agree on whether the natural world is all there is to know. (See my own article on Origins VIII 4 - Reflections on Religion and Big History) It simply ensures that we have common ground for our exploration and development of big history.
Finally, I think it is a rather bad idea to mix big history with religion. Religion cannot serve as a way of knowing when within any particular religious belief system it is not allowed to question the underlying assumptions. Take God for instance: that is just as much an assumption as the methodological naturalism is an assumption of science. Only that God is an assumption that empirically can neither be proven nor disproven. So, is this really a reasonable assumption to make when our aim is to work towards making the unknown known, to fill in the knowledge gaps? Bringing god into the discussion to me seems more like a way to end the discussion, or get lost in trying to figure out what is meant with god.
Also, trying to include religious perspectives effectively means trying to exclude non-believers. This is what religions have always done in the past, and not only religious belief systems, also secular belief systems. After all, it is only consequent when ones major aim is to uphold a belief system based on assumptions that are not to be questioned ever!
Science/academics should work as a dogma-prevention machine. Questions, investigations, formulating hypotheses and testing them against reality are the core of the scientific method. Belief systems defend themselves against all this by revolving around a dogma making that absolute. In a constantly changing world, this sooner or later clashes with reality.
Since this post is about academic versus experiential ways of knowing - I'll throw in a bit of personal experience here: I grew up in the former GDR/East-Germany and was properly indoctrinated (meaning I did believe it) into the belief system of Marxism-Leninism which aimed at installing Communism, but ended up in what we called "real-existing socialism" and there was no way you could question certain things or they would have locked you up. As simple as that. After the change (I was only 17 when the Berlin wall came down) I began to learn all sorts of things which I had no access to before and soon enough I started to think about religion as well. In fact, many of my relatives are believing Christians, but my parents had decided not to send us to religious instruction. I began to see the similarities between these two belief systems, only that the religious one is anchored in the idea of the supernatural and the other is just based on a grand idea of equality. Later on I discovered German thinkers who did the same, only professionally. One outstanding example is Joachim Kahl (Marburg) who left the church after having studied theology (he simply could not go on believing the way he did before) and then, as quite a number of people in the 1960 in West-Germany, instead started to believe in Marxism. Only after the change he realised that this is just another belief system that runs counter to reality and consequently said good bye a second time.
The answer, of course, is not to fall for an absolute relativism, meaning it is reasonable to assume that we can know about the world, and even if that knowledge is uncertain, it still is sufficiently certain to live our life.* It simply means that we have to become aware of the assumptions underlying our theories, ask ourselves if they are still reasonable to make and keep on working towards making the unknown known. We are doing ourselves a great disservice when instead we choose to simply declare the unknown to be the work of God (whatever that concept means - I still have not slightest idea). Is that what you mean with god of the gaps, Karen?
*and by the way, this amazing technology that allows us to communicate across the Atlantic is a very convincing proof of our knowledge of the world - it is not perfect, but it doesn't have to be. Just as much as we don't have to be perfect, we just need to do our imperfect best.