You are clearly on the right track.
In NY there is recent caselaw covering a lack of probable cause where the only allegation is crossing a fog line. I do NOT know what the state of the law is in NJ.
You might consider hiring a lawyer who knows how to phrase the motion; you may end up making unintended admissions.
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You have the right idea, and there is caselaw in New Jersey that addresses that specific fact pattern. The specifics of how far the car drifted, how many times, and other facts will determine the outcome. Filing such a motion is easy. Asking the right questions and winning the motion is a lot harder. Call me to discuss the case further.
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Erratic driving observed by law enforcement may give rise to a police stop under the community caretaking doctrine. You will need video of the 1 mile stretch of road that you mention. You will also need an attorney to file and argue your motion to suppress.
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You would have a right to file a motion and to get a hearing before a judge. This hearing would determine if the officer had the right to pull over your vehicle. The officer would be subject to cross examination by you or your attorney. This can be a very powerful tool to try to get your case dismissed. You would have the right but not the obligation to testify at the hearing. Clearly, you could attempt to introduce into evidence photos of the location to impeach (show to the judge that the officer is mistaken or less than truthful) in his testimony.
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