I'm a couple months late to the party, but sounds like OP was a hugely irresponsible pet owner. House too small for kid & a dog? Then you needed to either think more carefully about having a child, or rehome the dog. Your wife fucked up by not supervising the toddler and the dog more carefully. Sorry, but when kids are that small, you can't leave them unsupervised with a dog for a minute, let alone while you're doing something that requires concentration - cooking dinner.
Agree with the other posters that the dog was warning the child, and nipped at it without actually biting it. Which means that the dog didn't just haul off and bite the kid, it tried to tell the kid to stop whatever it was doing.
A pet is not a human, it's true. But it is a life form, one that you have made the conscious decision to care for. It is self-aware, it is intelligent, it has feelings and emotions, and it depended on you for care. Your poor dog. Fourteen years thrown out the window because of human error. I call that a tragedy.
I'm so with Blaise Pascal in that "[t]he more I see of Mankind, the more I prefer my dog." The irony of this quote should not be lost on anyone.
Don't get any more pets. Seriously. If this wasn't lesson enough for you that you cannot responsibly care for a pet, then I don't know what to tell you.