Accuray
I understand that your kids being involved changes the equation considerably. That being said your wife handed you a major fitness test and you failed it. You had an honorable reason to let it go, unfortunately you just proved to her that she can say or do outrageous things in front of the children, and you will fold for their sake. Don't be surprised if she continues to throw ultimatums and tantrums in front of them.

I know it seems like you are making things worse, but when she throws a tantrum let her. Let her learn that she gets nothing out of it. Otherwise you are just reinforcing to her that she can get her way if she acts nasty in front of the kids.