Okay, here is the thing: what happens when you invite Billy Smith and Family and he has 7 siblings?
But, if you are going to invite the entire family for real... how formal is the party? At a party for a 6 year old, which I would expect to be very casual and child-oriented, I probably would not address my invitations formally, at which point "Child and Family", to me, is cute. No, it's not proper etiquette but you're probably not serving soup, a fish course, a meat course, salad and cheese course, a dessert course, and a fruit course.
Also, it sounds like you are stuck with a large party because of your own family, and that adds to the stress of extra kids without THEIR parents. You might consider having a family party separately, and then letting your child invite over a few friends on another day. That way you can limit it to a manageable number without the pressure of adding the entire family of each friend you invite. Just a thought.