Sorry Op, I know that you are trying to do good thing. But fighting fire with fire is not always the best option.
You know that you shouldn't fight with trolls and internet morons, the best course of action is to avoid and ignore them.
I hope you understand that your action might create more harm that good.
IMHO, the best solution is to post explanation on their Facebook - similar to what you told us in op and every sensible person should understand.
Neither of the two options is actually good, because posting explanation on their fb cant stop some that defy all logic when it comes to stereotypes.
And flooding the thing with replies that are obv. fake wont do any good either.. Hard to be smart in this position.