Mrs. Merriweather does make her case that the missionary circle and the "saintly" J. Grimes Everett are fighting the good fight, but the good fight is converting the Mrunas to Christianity because they supposedly live in sin. She might be sincere in that she does support non-Christians being converted to Christianity. And she might be sincere in supporting Everett in that cause, but she would certainly not fight the fight with Everett. She is sincere only in keeping her distance from people like the Mrunas: people of African descent.
In short, are the missionary ladies (sans Miss Maudie and maybe Aunt Alexandra) sincere in worrying about the well-being of the Mrunas? The answer is absolutely not. Mrs. Merriweather and Mrs. Farrow in particular are complete hypocrites. If they really supported the Mrunas and other non-white people, they would not talk so disparagingly of African-Americans in their own country.
For instance, to voice her opinion that "they" (black people) are still dangerous, despite being given religious instruction, Mrs. Farrow adds, "We can educate ‘em till we’re blue in the face, we can try till we drop to make Christians out of ’em, but there’s no lady safe in her bed these nights."
Mrs. Merriweather is perhaps the biggest hypocrite of them all. The so-called most devout lady in Maycomb, Mrs. Merriweather preaches her support for the poor Mrunas (a condescending support at best), but she treats her maid Sophy like a second class citizen who has the mind of a child. In the wake of Tom's trial, Mrs. Merriweather notes that she told Sophy to stop grumbling about it and act more like a Christian. Mrs. Merriweather also goes on to criticize Atticus (although she never mentions his name) for stirring up hope (for African-Americans in town) by serving as Tom's lawyer. To top it off, Mrs. Merriweather calls the people in the North hypocrites because they fought (in the Civil War) to free the slaves. She notes:
People up there set ‘em free, but you don’t see ’em settin‘ at the table with ’em. At least we don’t have the deceit to say to ‘em yes you’re as good as we are but stay away from us. Down here we just say you live your way and we’ll live ours.
Mrs. Merriweather is not sincere in truly caring about the Mrunas. She clearly thinks whites and blacks are not equals. In that last statement, she implies that she thinks white people are superior. "At least we don't have the deceit to say to 'em yes you're as good as we are . . . " And notice that she tolerates black people in town as long as they stay in separate places from whites. If she and her missionary followers actually worry about the Mrunas, she does so in a condescending way, the kind of worrying that the Mrunas live decent lives as long as they are declared to be inferior to, and separate from, whites.