Should I fight a seat belt citation or pay and take traffic school?

Departing from the Publix store I got into the passenger seat of my brother's car. As soon as I jumped in, my brother began to depart. I delayed putting my seat belt on as I noticed that he had given me my paycheck and I decided to put it into my wallet that was in a zipped pocket on the side of my jacket before attaching the seat belt. My brother turned out of Publix and onto Williamson Blvd and before we reached the red light (200 ft) I had completely buckled in my belt. We sat at the light for 20 or 30 seconds and after we crossed the intersection a police car behind us flashed its lights and when my brother Bernard said "What did I do" that is when I realized there was a police car behind. I had never seen the car and definitely did not put my belt on because I might have seen the car. Never did. I am 60 years and have ALWAYS worn my seat belt. Upon the stop the officer said that he saw me putting on my belt. Later on the next day I found out that he was in training, and also had incorrectly marked down the wrong statute on the citation (I called him with that info so that he could correct it yesterday).He never questioned me about the incident and I was very passive and did not question him. My question is that since this is a non-moving violation should I just pay the fine and go to traffic school to get the "adjudication withheld"? I have never been in an accident, never issued a traffic citation in my life, and I know that in this case I was late in putting on my seat belt but I was putting it on and not because I saw him. I would take a lie detector test to prove that. The officer must have seen us coming from the Publix so a little common sense, not like we were driving for a distance with no seat belt, but I realize chances of prevailing are slim.