don't cancel your teaching gig,
find someone to pick up the kids on that day and then work out some arrangements with someone else to help you with that, if it's only for a couple of hours, someone will be able to help, I'm sure of it, SP you're a smart guy you can make it work. As for your wife's behavior, let her know the less time she spends with the kids that you'll be seeking more support from her - if she wants to make it harder on you, you are always free to return the favor ;-)