بسم الله الرحمن الرحيم
A woman is allowed to delay performing hajj due to minor child but if she is of sound health and able to perform hajj then it is not lawful to ask someone to perform Hajj-e-Badal on her behalf. Hence in the question mentioned above when your wife is alive and healthy and Hajj has become fardh on her and she herself is able to perform Hajj then merely due to the child it shall not be lawful for you to offer Hajj-e-Badal on her behalf. Yes if there is some problem regarding the care of the child then it is allowable to delay the hajj.
والوالد الصغير المحتاج إليه عذر في التخلف مريضاً كان أو لم يكن . (غنية الناسب، ص: 10، مقدمة في تعريف الحج و ما يتعلق به ، ط: مكتبه يادگار شيخ)
Allah knows Best!
Darul Uloom Deoband