Honestly I don't think putting a 6-year-old with an "insufferable, incompetent, unsympathetic teacher" every day, all day, all year long is going to teach him resilience. Just think about how may hours of his life he's spending with this person- like it or not, she has a lot of influence over him. If you think the influence is more bad than good, I would take him out of the class.
If he were a teen my advice would be different, particularly because by high school he'd likely only have to endure a bad teacher for one hour a day.
And just for perspective my 6 year old probably cries three times an hour so three in six months is nothing.
