If you ever prove customers mistake. they will never come to buy your service again.
customers mentality is that he is always right.
and when they pay 1$ item from your company they think they have bought the entire company so they can express whatever the crap they have into their minds.
sad reality but you have to treat your customer like wife treats to her husband. you have to listen to him even he is not on right.
and there is no solution for this because once you started being logical they the entire culture will go against you.
so the best part is just avoid stupid clients and always find smart people to work with.