I apologize for the confusion! We have taken a second look at this and have found that we were too hasty in declaring this an error: ultimately, we've determined that the second werden is in fact optional, rather than mandatory.
To explain our reasoning in full (as it may be useful to know when speaking with some native speakers):
Essentially, what is happening is that würdest is the Konjunktiv II Präteritum of werden, AND, in colloquial speech, it can be used as a modal verb to turn any verb into Konjunktiv II - this latter format is normally used for verbs that don't have a Konjunktiv II Präteritum form that is colloquially used (e.g. essen - you would normally say ich würde essen instead of ich äße).
For verbs that do have a common Konjunktiv II Präteritum form, like haben, you would normally say ich hätte, but you could also say ich würde haben to achieve the same meaning. It isn't necessary, however, and it's better to stick with the most common form, which in this case would be hätte. So this is the same thing that has happened on our end here: some native speakers might say ich würde werden to put werden into Konjunktiv II, but this isn't necessary: ich würde has the exact same meaning. This is how you may hear a native speaker comfortably say something like ich würde geschnappt werden - to their ears, this sounds like the accurate and complete passive form. Grammatically, though, the werden is redundant.
So, ultimately, the original sentence, Spätestens am Flughafen würdest du dann aber doch geschnappt, is indeed grammatically correct, and the werden isn't actually necessary. As you had suspected, the passive is already clearly indicated by the use of würdest + geschnappt.
My apologies once again for causing you confusion and for our initial error! I hope that this clears everything up.