The problem (or, according to some posters, one of the problems) with your idea is that the concept of a polynomial-time algorithm is itself a mathematical question and not a question about reality. If I gave you the running times of an algorithm for input sizes from 1 to 3^^^3, this would not tell you whether the algorithm is polynomial. However, it would answer the question as far as reality is concerned, since it's impossible to perform the algorithm on inputs bigger than that (or even remotely close to that size).
For simple algorithms like bubble sort or whatnot, we can look at the algorithm and say, "oh, that's quadratic time" and that lets us make predictions about how long it will take to run (if it takes 10 seconds to sort a 10^9 element array, then it will probably take 1000 seconds to sort a 10^10 element array). But this is not a statement about reality directly -- this is a mathematical statement which describes reality, in this case.
Furthermore, it is impossible to make this sort of statement in general. Just as we can't tell if an arbitrary program halts or not, we cannot in general say if an arbitrary algorithm runs in polynomial time. In fact, I'd bet that one can construct an algorithm such that the question whether the algorithm runs in polynomial time is independent of ZFC. This is not a question of ZFC failing to describe reality, because it's not a question about reality whether the algorithm is polynomial, to begin with.
As for P=NP... there's actually an interesting exercise that I'd expect anyone in a good theoretical CS class to see. Right now, without doing cutting-edge research, I can write down an algorithm for solving 3SAT with the following property: if P=NP, then the algorithm I write down will run in polynomial time (and obviously the converse also holds). Thus, deciding whether P=NP is isomorphic to determining the running time of this algorithm (which has a very simple description!) As I've stated earlier, it would not be too surprising if this question turned out to be independent of ZFC.
Edit: it appears that my last paragraph has been preempted.
Also: upvoted for an interesting question.
Suppose we had a model M that we thought described cannons and cannon balls. M consists of a set of mathematical assertions about cannons, and the hypothesis is that these fully describe cannons in the sense that any question about cannons ("what trajectory do cannon balls follow for certain firing angles?", "Which angle should we pick to hit a certain target?") can be answered by deriving statements from M. Suppose further that M is specified in a certain mathematical system called A, consisting of axioms A1...An.
Now there is much to be said about good ways to find out whether M is true of cannons or not, but consider just this particular (strange) outcome: Suppose we discover that a crucial question about cannons - e.g. Q="Do cannon balls always land on the ground, for all firing angles?" - turned out to be not just un-answerable by our model M but formally independent of the mathematical system A in the sense that the addition of some axiom A0 implies Q, while the addition of its negation, ~A0, implies ~Q.
What would this say about our model for cannons? Let's suppose that we can take Q as a prima facie substantive question with a definitive yes or no answer regardless of any model or axiomatization. At the very least it seems that M must be an incomplete model of cannons if the system in which it is specified is insufficient to answer the various questions of interest. It seems to me that
If a question about reality turns out to be logically independent of a model M, then M is not a complete model of reality.
Now we have an axiomatization of mathematics -- let's say it's ZFC for now -- and we have a model of computation in reality, which is M="The unvierse can contain machines that (efficiently) compute F iff there exists a Turing machine that (efficiently) computes F" with appropriate definitions of what exactly a Turing machine is in terms of ZFC. Suppose we want to answer a question like Q="Can the universe contain machines that efficiently solve SAT?"
Under the premise that M is true, the question Q becomes the pure logical question R="Are there Turing machines that efficiently solve SAT?", i.e. the P versus NP problem.
Now suppose that R was shown to be formally independent of ZFC in the sense that for some axiom A0, ZFC+A0 implies P=NP and ZFC+~A implies P!=NP. This would resolve the mathematical question of P versus NP but the question Q seems like a prima facie concrete question with a definitive yes or no answer that does not rely for its substance on M or ZFC or any other epistemic construct. It would seem that we must have missed something in our description of reality, M.
Perhaps more controversially, I claim: Under the correct model M' it seems that it's impossible for a substantive question (such as Q) to be unanswerable.
All this adds up to: The P versus NP problem (and questions like it that can be phrased as definitive questions about reality) must have an answer unless our model of reality is incomplete.