If nature, or any aspects of it, are intelligently designed, how could we tell? How would we be able to test that hypothesis?
Only someone who believes in intelligent design can answer that question for you and that answer will most likely be unscientific (therefore not fit for this site). The reason is because intelligent design is a claim that is not based on observable reality rather divine revelation so it is impossible test any hypothesis regarding it.
Evolution on the other hand is based on two observable, testable facts: (1) all living things on Earth inherit their traits from their parents and (2) those traits are not always copied and passed on with perfect fidelity. With these two facts we can theorize on what other observable facts lead to the variation in life that we see today i.e. evolution by natural selection.
I have never read any testable hypothesis from the Intelligent Design (ID) community. As such ID is not wrong or right, it is just not testable.
To be fair, ID contains a lot of logical fallacies. Not only does it not offer any testable hypotheses but it often just does not make much sense anyway!