Trying to make a usable theory that also proves its own consistency is probably also a futile goal, per Gödel, but it's a separate one, and not one that people would likely go for if an inconsistency were found in Peano arithmetic.
Are you sure it isn't the other way around? If you can prove a contradiction using intuitionistic logic, it's also a contradiction using classical.
If you proved the contradiction using the excluded middle then that proof wouldn't hold when you use intuitionistic logic. Maybe I'm missing or not understanding something here.
What's surprising is that in this particular case (the Peano axioms), the reverse is also true; you won't be able to eliminate contradictions by passing to this weaker logic.
Note, all I said is, if you can prove a contradiction classically, then you can prove a contradiction constructively! I didn't say, if you have a classical proof of a contradiction then it's a valid constructive proof of a contradiction. Obviously not! But it will still be true that you'll be able to prove a contradiction constructively; it just won't necessarily be the same proof.
If you want to know more about the particular transformation, well, here's a relevant Wikipedia article: https://en.wikipedia.org/wiki/Double-negation_translation
I know this is off-topic but if you have a book recommendation which teaches about intuitionistic logic/constructive math and also mentions the double negation translation, I'm interested. Most books I've read seem to either concentrate on only constructive or classical math.
[0]: https://en.wikipedia.org/wiki/Double-negation_translation
Above I said the only reasonable way to weaken Peano arithmetic was to weaken induction. But that's not quite true. There's another way: Not changing any of the axioms, but weakening the underlying logic.
See, there's a school of mathematics known as "constructivism", and they object to the usual laws of logic, saying that they're too strong; they use a weaker logic. It ought to be called "constructive logic", but for historical reasons (that are really not worth getting into) it's called "intuitionistic logic" instead.
And this might seem to be a better approach, because weakening the laws of logic is something that can be done without regard for the setting; it doesn't matter here whether you're talking about just whole numbers, or mathematics as a whole. (Well, almost. Actually, from ZFC and intuitionistic logic, you can effectively get back classical logic. But there are pretty good ideas about how to modify ZFC to avoid that.)
The problem is, doing this doesn't help. Because it's known that if you can prove a contradiction from the Peano axioms using classical logic, you can also do so using intuitionistic logic. So even if you went constructive, you'd still be stuck with all the same problems.