There isn't a unified "string theory controversy".

The battle-tested part of fundamental physics consists of one big intricate quantum field theory (the standard model, with all the quarks, leptons etc) and one non-quantum theory of gravity (general relativity). To go deeper, one wishes to explain the properties of the standard model (why those particles and those forces, why various "accidental symmetries" etc), and also to find a quantum theory of gravity. String theory is supposed to do both of these, but it also gets attacked on both fronts.

Rather than producing a unique prediction for the geometry of the extra dimensions, leading to unique and thus sharply falsifiable predictions for the particles and forces, present-day string theory can be defined on an enormous, possibly infinite number of backgrounds. And even with this enormous range of vacua to choose from, it's still considered an achievement just to find something with a qualitative resemblance to the standard model. Computing e.g. the exact mass of the "electron" in one of these stringy standard models is still out of reach.

Here is a random example of a relatively recent work of string phenomenology, to give you an idea of what is considered progress. The abstract starts by saying that certain vacua are known which give rise to "the exact MSSM spectrum". The MSSM is the standard model plus minimal supersymmetry. Then they point out that these vacua will also have to have an extra electromagnetism-like force ("gauged U(1)_B-L"). We don't see such a force, so therefore the "B-L" photons must be heavy, and the gist of the paper is to point out that this can be achieved if one of the neutrino superpartners acts like a Higgs field (by "acquiring a vacuum expectation value"). In fact this paper doesn't contain string calculations per se; it's an argument at the level of quantum field theory, that the field-theory limit of these string models is potentially consistent with experiment.

That might not sound exciting, but in fact it's characteristic, not just of string phenomenology, but of theoretical particle physics in general. Progress is incremental. Grand unified theories don't explain the masses of the particles, but they can explain the charges. String theory hasn't yet explained the masses, but it has the potential to do so, in that they will be set by the stabilized size and shape of the extra dimensions. The topology of the extra dimensions is (currently) a model-building choice, but once that choice is made, the masses should follow, they're not free parameters as in field theory.

As for what might determine the topology of the extra dimensions, anthropic selection is a popular answer these days - and that has become another source of dissatisfaction for string theory's critics, because it looks like another step back from predictivity. Except in very special cases like the cosmological constant, where a large value makes any kind of physical structure impossible, there's enormous scope for handwaving explanations here... Actually, there are arguments that the different vacua of the "landscape" should be connected by quantum tunneling, so the vacuum we are in may be a long-lived metastable vacuum arrived at after many transitions in the primordial universe. But even if that's true, it doesn't tell you whether the number of metastable minima in the landscape is one or a googol. This is an aspect of string theory which is even harder than calculating the particle masses in a particular vacuum, judging by the amount of attention it gets. The empirical side of string theory is still dominated by incrementally refining the level of qualitative approximation to the standard model (including the standard cosmological model, "lambda CDM") that is possible.

As for quantum gravity, the situation is somewhat different. String theory offers a particular solution to the problems of quantum gravity, like accounting for black hole entropy, preserving unitarity during Hawking evaporation, and making graviton behavior calculable. I'd say it is technically far ahead of any rival quantum gravity theory, but none of that stuff is observable. So approaches to quantum gravity which are much less impressive, but also much simpler, continue to have supporters.

I don't do formal Bayes or Kolmogorov on a daily basis; in particle physics Bayes usually appears in deriving confidence limits. Still, I'm reasonably familiar with the formalism. As for string theory, my jest in the OP is quite accurate: I dunno nuffin'. I do have some friends who do string-theoretical calculations, but I've never been able to shake out an answer to the question of what, exactly, they're calculating. My basic view of string theory has remained unchanged for several years: Come back when you have experimental predictions in an energy or luminosity range we'll actually reach in the next decade or two. Kthxbye.

The controversy is, I suppose, that there's a bunch of very excited theorists who have found all these problems they can sic their grad students on, problems which are hard enough to be interesting but still solvable in a few years of work; but they haven't found any way of making, y'know, actual predictions of what will happen in current or planned experiments if their theory is correct. So the question is, is this a waste of perfectly good brains that ought to be doing something useful? The answer seems to me to be a value judgement, so I don't think you can resolve it at a glance.

What on earth is the String Theory controversy about, and is it resolvable at a glance like QM's MWI?

I wonder how you resolve the MWI "at a glance". There are strong opinions on both sides, and no convincing (to the other side) argument to resolve the disagreement. (This statement is an indisputable experimental fact.) If you mean that you are convinced by the arguments from your own camp, then I doubt that it counts as a resolution.

Also, the Occam's razor is nearly always used by physicists informally, not calculationally (partly because Kolmogorov complexity is not computable).

As for the string theory, I don't know how to use Bayes to evaluate it. On one hand, this model gives some hope of eventually finding something workable, since it provided a number of tantalizing hints, such as the holographic principle and various dualities. On the other hand, every testable prediction it has ever made has been successfully falsified. Unfortunately, there are few other competing theories. My guess is that if something better comes along, it will yield the string theory in some approximation.