I don't think you're taking the thought experiment seriously enough and are prematurely considering it (dis)solved by giving a Clever solution. E.g.

If it's not the urge, what is it?

Obvious alternative that occurred to me in <5 seconds: It's not the urge, it's the actual act of smoking or knowing one has smoked. Even if these turn out not to not quite work, you don't show any sign of having even thought of them, which I would not expect if you were seriously engaging with the problem looking for a reduction that does not leave us feeling confused.

Edit: In fact, James already effectively said 'the act of smoking' in the comment to which you were replying!

becomes interesting if even after accounting for the urge to smoke, whether you actually smoke provides information on whether you are likely to get lung cancer.