Neither of you has enough miles to get one single award ticket in economy from US to Australia. Even if you did, award tickets are (generally) non-upgradeable.

Meaning you will have to buy 3 tickets, let it be economy or business. It seems you prefer to fly business. Keep in mind, the ticket (in business) for your child will not be cheap. What's your budget?

Using miles to upgrade your (bought) economy tickets to business is kind of possible, if the fare is upgradeable (in other words, more expensive) and seats available (without elite status, chances are slim between US-Australia). I don't think you currently have enough miles to upgrade 3 round-trip tickets though.