this is an exceptionally well written, and more importantly, well argued essay that brings a novel, and intriguing, perspective to the question of whether non-computable objects matter in the world.
It's an interesting perspective to try to accord an abstract quantity causal significance. On the face of it, it seems to invite some strange notions---if I say, 'the real number line contains more elements than the set of natural numbers', has 'the real number line' with its uncountably many elements caused me to say that, and if so, do we need to say that it exists, in some way? What about 'the round square copula on Berkeley college, which is pink'? This invites the issue referred to by Quine as 'Plato's beard'---and dealt with by his dictum 'to be is to be the value of a bound variable', so that the offending object can be freed from any causal relevance---'there is something, such that it is round, square, pink, and the cupola of Berkeley college' does not need for there to be such an object, in order for me to utter this sentence. Hence, it's not that what's caused me to utter the phrase.
But of course, you follow a different approach. Abstract objects, in your case, exist because they have causal consequences regarding what happens in the world, not merely what is being referred to. One can't assert the reality of Pegasus just by referring to it, but if Bellerophon is carried away by a winged horse, it would be hard to deny its existence. So you say, odd numbers make it hard to split a check; electric fields move particles around. Hence, these things exist.
I'm of two minds regarding this argument. First of all, it seems to me that there can be, strictly speaking, inconsistent mathematical theories that nevertheless successfully describe physical situations---quantum field theory and general relativity are mutually inconsistent, yet can be brought together to predict, say, black hole evaporation. Certainly, we expect for there to be a further theory such that it is consistent, and reduces to the others in the appropriate limits, keeping its predictive successes, so that one can say, well, the objects of that theory exist, instead; but it at least urges caution with respect to this sort of argument.
However, I don't think that your argument is suspect, in this sense. (It's also a thing of beauty, I have to say.) Essentially, you appeal to the fact that the algorithmic mutual information, in the average, approaches the Shannon mutual information only if the initial probability distribution is of low KC---and thus, if we find that whenever things are correlated (mutual information measures correlation), they are also co-explainable, we obtain evidence for the smallness of the initial probability distribution's KC. In this sense, Kolmogorov complexity has a hand in the makeup of our world; no being with access only to computational resources could have set things up this way. If god exists, she must have access to an oracle.
As I said, I think this is a highly ingenious argument. But isn't it vulnerable to anthropic counterarguments? It's easy, given sufficient resources, to simply compute all universes. Then, there will be some with low initial Kolmogorov complexity; only in those will there be beings that have had any luck at explaining their world, and thus, that could hold some sort of conversation like the one we're having here. But if the world were like that, the conclusion that KC plays some fundamental role would be misguided---the appearance would be an accident of placement, so to speak.
Anyway, I'm running out of time, and I haven't even gotten to the part I liked the most---namely, your notion of different explanatory reference frames. If I read you correctly, one way to frame this is that different explanations make sense to different people, because, essentially, while both explanations may be correct, one could take an unfeasibly long time to unpack for person A, with the issue reversed for person B---so even though both, within limits, are saying something true (although one explanation might be the more simple one, we couldn't really show this to be the case), they might be mutually almost unintelligible.
I think this is a highly intriguing thought, and deserves to be developed further. So, I wish you the best of luck in this contest, thanks for submitting your essay, I will spend some more time thinking about the issues it proposes!