Things equal to the same thing are equal to each other; if 4=1, and 2+2=4, then 2+2=1.
Yes. But it still equals 4. If you are trying to demonstrate a case where 2+2 isn't 4, this isn't it. You've demonstrated a difference, just not the difference you say you demonstrated. Hence, your original example was bad.
Ah. What does it mean to say that the integers (not mod 3) exist, if it is an observed fact that 2+2=1? That is, that what we think of as an integer number of objects is actually an integer mod 3? If I dispute your assertion, and say that the integers (not mod 3) do not actually exist, what experimental outcome will demonstrate that I am wrong?
There's are a few problems with your question. Firstly, when we ordinarily say "2+2!=1", this is implicitly a statement about integers, not integers mod 3. That is to say, "2+2!=1" isn't a complete statement at all; it only makes sense to the extent that we know what "2", "+", and "1" indicate. As a statement about integers, with the usual interpretations of thse symbols, it's true. As a statement about integers mod 3, with the usual interpretations of these symbols, it's false. It's not one statement being true in one universe and false in another, it's just two different statements.
Secondly, numbers are models. Remember, the universe does not run at the level of "objects", it runs at the level of particle fields (or something like that). We count objects using whatever system of numbers is appropriate for counting objects (to the extent that "object" is a sensible notion). But even if whole numbers were somehow not a good model for counting objects, there are still plenty of other things that they would be a good model for, and they come up pretty naturally mathematically even without that.
So aside from a proof of the inconsistency of mathematics, not much could convince me that the whole numbers don't exist. There are plenty of experimental outcomes, however, that could convince me that they aren't a good model for something we're currently using them for (e.g. counting objects, if 4 objects were indeed the same thing as 1 object, whatever that means).
This is intended to explore a a thought I had, rather than making any particular argument about truth.
The canonical example of a thing which is true without any obvious physical referent is the statement 2+2=4. It is true about fingers, sheep, particles, and galaxies; but intuitively it does not seem that any of those truths encapsulates the full meaning of the statement. Moreover, it certainly seems that there is nothing anyone could do to make the statement untrue; it seems that it would have to hold in any universe whatsoever.
Now my thought: How do we know that the physical universe operates on this sort of arithmetic, and not arithmetic modulo some obscenely large number? Suppose we repeat the experiment that convinces us 2+2=4 (and let's note that babies are presumably not born knowing this; they learn it by counting on their fingers, even if they do so at too young an age to express it in words), but with much larger integers. Perhaps we might find that, when we take 3^^^^3 particles, and add 1, we are left with 3^^^^3 particles without any awareness that any particles have disappeared. And what is more, if we take three sets of 3^^^^3 particles, and measure their mass separately and then together, we find that we get the same mass. After some long sequence of such experiments, perhaps we might convince ourselves that physics actually operates on integer arithmetic modulo 3^^^^3. (Which would be unexpected in that the physics we know operates on complex numbers, not integers, but perhaps that's an approximation to some fantastically-finegrained two-dimensional integer grid.)
What would this mean, if anything, for the truth of such statements as 2+2=4? It seems that it would then be a contingent truth, not a universal one; that there could in principle exist a universe whose physics operated on arithmetic modulo 3, so that 2+2=1. (Presumably such a universe would not have any sentient beings in it.) What if 2+2=4 is an observed fact about our universe on the same order as the electromagnetic constant or the speed of light?