If you find a right triangle with sides (2.9, 4, 5.15) rather than (3,4,5), are you ever entitled to reject the Pythagrean theorem?

That is not what I said. I said, regarding Pi == 3, "this won't tell you the actual value of Pi, nor would it constitute a mathematical proof, but at least the experience would point you in the right direction". If you believe that a^2 + b^2 = c^5, instead of c^2; and if your instruments are accurate down to 0.2 units, then you can discover very quickly that your formula is most probably wrong. You won't know which answer is right (though you could make a very good guess, by taking more measurements), but you will have enough evidence to doubt your theorem.

The words "most probably" in the above sentence are very important. No amount of empirical measurements will constitute a 100% logically consistent mathematical proof. But if your goal is to figure out how the length of the hypotenuse relates to the lengths of the two sides, then you are not limited to total ignorance or total knowledge, with nothing in between. You can make educated guesses. Yes, you could also get there by pure reason alone, and sometimes that approach works best; but that doesn't mean that you cannot, in principle, use empirical evidence to find the right path.

Peer review. If the next two hundred scientists who measure your triangle get the same measurements from other rulers by different manufacturers, you'd be completely justified in rejecting the Pythagorean theorem.

My challenge to you: go out and see if you can find a right triangle with those measurements.