To answer your question, no, you cannot get the ticket dropped. The justification of the stop, and penalty are legit.
If something was found during the Trooper's search, and you were dinged with it, THEN you'd have something to have "dropped."
Now in the purest sense, your rights were violated. If you wished to remedy the infringement, you would be justified in bringing forth a suit.
But that's about it.
This right here.
Two different issues...