Escalate through twitter team. While you may not be technically entitled to IDB compensation, I have seen AA use the same amount for other accidental denied boarding incidents.

You should also feel pretty safe in refusing to de board. Everyone has smart phones, the PR nightmare of having the cops come on and forcibly remove a family with an infant is not something any airline wants to risk in the post-dr dao era. Once your seated with a valid ticket/BP the only way I'd be vacating my seat is if they offered enough compensation/rebooking that I voluntarily left.