How would one begin to try to identify the properties of the mathematical structure corresponding to physical reality? I was thinking about the Mathematical universe hypothesis and a natural question popped into my mind:
Assuming that the universe (by universe I mean the complete physical reality) is really isomorphic to some conceivable, mathematically constructible structure, how would one begin to narrow down the possibilities? How would one identify its properties without necessarily finding the structure itself?
My first guess is that one should look at QFT and GR and assume that the mathematical structure we want would have to be consistent with those two theories at least in the appropriate approximations/limits and that we could somehow find all the familiar symmetries in some form on that structure.
But those are just words, I don't understand what would one have to do rigorously to rule out some of the structures, I would really appreciate some non-handwavy guidelines if it's possible.
P.S. I'll gladly elaborate further and edit my question if something is particularly unclear.
EDIT 1: I'm not asking how theoretical physics should proceed in general and how should the scientific method be used in order to understand the world. I'm interested in how much can we say about the "final, true and complete" theory (assuming it exists) without actually having it. I'm interested in the mathematical structure associated with that theory, what are the most general statements about it that are almost certainly true?
 A: If I had to describe physics in one sentence it would be "Ruling out mathematical models and finding predictive new ones that are compatible with previously established ones". You seem to be misunderstanding what the "Mathematical universe hypothesis" is. It's a mostly philosophical notion that any mathematical structure, satisfying certain conditions, can be considered "reality". The idea is an elaboration on, not an invention of, the use of mathematical modeling in physics. Physics has always progressed very much analogously to what you yourself have described when you said :

My first guess is that one should look at QFT and GR and assume that the mathematical structure we want would have to be consistent with those two theories at least in the appropriate approximations/limits and that we could somehow find all the familiar symmetries in some form on that structure.

We've always done this with whatever theories we have at hand in the hopes of moving on to newer and broader ones. You've basically answered your own question here. It's hard to get more specific because your question addresses a broad issue and a more specific answer would depend on the specific context. While none of these theories are "true and complete' (as you call it) they are the closest we can get to an accurate representation of reality, despite ultimately being "wrong" in the absolute sense of the word.
Indeed, you also seem to give great importance to what you call "true and complete" theory of physics beyond what we can ascertain using observational predictions. From a scientific perspective, this is meaningless. There is no need to assume our theories are "true" in the absolute sense of the term. Physical theories are evaluated by their experimental usefulness. This does not guarantee any truth value to them in the absolute sense, they are "merely" extremely useful representations of reality. Again, the key point here is that these representations are only judged by their observational and experimental usefulness. Asking about the absolute truth, as you seem to be, beyond representations that we can validate through experiments, lies outside the realm of physics. Because the only value judgement a scientist can ever offer to an idea is through its experimental usefulness, un-testable ideas become meaningless in this perspective.
A: First, a distinction between model of a system and the system. A model can be an approximation (up to a degree) or accurate (either to whole system or part of it in space and time) or The map is not the territory (in a post-modern twist :) ).
A mathematical model is just a description which uses mathematical language, one can very well use another desription.
As one philosopher used to say, the fact that water has this representation/description ($H_2O$) is not necesarily better description than saying that water boils in $100^o$ celsius.
R Feynman used to say how his father talked about birds (not exact quote in "Genius: biography of Richard Feynman"):

..people use to name things, say this bird is called such and such
  etc.. but the main point is to know what the bird does and how it
  functions and not just a name for it..

Mathematics is a language and as such can describe things, the fact remains that the description is based on the system (lets say reality) and observation/testing and not the other way around.
i find what Einstein said very good (not exact quote in "Einstein: Philosopher, Scientist")

the type of mathematics we do depends on the physical reality we live,
  since according to the physical reality some (geometric) theorems may
  be true or not etc.

A: The word you are looking for is called "experiment". It is the one and only means in science to approach nature. Mathematics is merely a handy tool to interpret experimental data. 
Physics students are usually told something rather profound about their craft in one of the early classes, usually the first theory class they take. My professor approached it this way:
"Physics,", he said, "is the art of approximation. If you can't live with this horrible thought, that none of what I am going to teach you here has any absolute quality, I would suggest that you get up now, and walk over to the philosophy department."
And after a short but diabolical smile indicating how little he thought of the philosophy department, he began teaching us the most successful art of approximation that humans know about. What he never taught us, was to mistake ANY mathematical models of reality for reality itself. 
As a corollary, not even the mathematicians think of mathematical structures as absolutes. Their existence depends very much on the choice of the logic system that one picks for the derivation of theorems from axioms. Many non-trivial mathematical structures exist only in one type of logic, but not in others. The fact that most mathematicians agree to work in one particular framework (which has a particularly rich set of consequences for algebraic and topological structures), should not deflect from the fact that that choice is both historical and arbitrary. 
