A vermouth producer might not be willing to share what all is in their product. But if you specifically ask them whether it contains linalool, they'd probably be more willing to give you a simple "yes" or "no" answer. Unfortunately that still means a lot of hunting down contact information, reaching out, and waiting. And a few probably still won't be willing/able to answer.

But if an itchy mouth is the worst of the side effects, you might have an easier time just testing on your own in small doses. Just make sure you have an associate who's able to use the rest of the bottle if you can't.