This is what i recommend. Fight it. The cop needs to show up in court to say it was valid and in a court, her word is as good as your word. ONLY because you say you have had a clear, completely, driving record. You really have a good chance. Hell it might HONESTLY be worth it to visit a lawyer, depends though.
1. pay a lawyer like 100-500 fro him to show up and the judge to let it go.
2. pay the court fine, possibly have insurance go up 30 PLUS a month for 3 years.. thats (30*36)=1080. (does depend greatly on age etc.)
And that "broad" statement was fairly, well no it was very ignorant.