You're proving his point. Discussion of Peano axioms or however you want to construct natural numbers is irrelevant to the question what's two plus two, which has a straightforward answer to anyone who isn't being intentionally obtuse.
A better point of comparison would not be a question like "what is 2+2," but "what is 4?" There's a superficial, circular answer ("What is 4? It's 2+2"), and there's a more complex and rigorous answer which doesn't look good on camera ("What is 4? Well, numbers are a social construct used to communicate and analyze etc etc, Peano arithmetic blah blah").
These questions also admit "straightforward" answers ("what's 4?" raises 4 fingers "this many") ("what's a woman?" points to a woman "one of those"), but these don't really answer the deeper question being asked. They gesture at a preexisting category and demand that it be recognized without actually explaining the nature of that category or its boundaries.
Brother, you'll never guess what type of construct numbers are.
If American voters prefer simple, incorrect answers over complex truths, that's a problem with their education system, not with trans rights.