Your wife already says that noone there likes her...so she doesn't want to go.
If you don't go, she will then say that she is mad at you for not going...because noone is going to like here >even worse< now because you didn't go...because of her. Plus...the kids are gonna be mad at her for a long time...casue they think she's faking it.
If you go, yes you are leaving your sick wife at home. (It is, however, NOT a debilitating sickness...its a minor infection...as long as by "stuff" in her urine they are not saying blood)
So...what I would do...is force #3 child to go with you...thus leaving the wife totally alone to recover.
Stated as..."Honey, I'll take all the kids...that way you can have a mini-vacation without us and you won't have to worry about the girls...you can relax and let the antibiotics do their work". I would much rather deal with an angry child than a spurned wife.
You'll be good as gold (eventually)