All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
You should keep the child's clothes and body pure, especially during Tawaaf as purity is one of the conditions of the validity of Tawaaf. The guardian should wrap the child so that the dirt would not spread through his clothing while he is carrying him. As for cutting the hair of the baby after assuming the state of Ihraam, it is impermissible as this is one of the prohibitions of Ihraam and it is the duty of the non-discerning child's guardian to make him avoid the prohibitions of Ihraam. If the guardian performed Tawaaf and Sa‘y with the child, he should cut his hair to end his state of Ihraam.
Shaykh Ibn Baaz mentioned what the guardian should do if he wanted to perform ‘Umrah for the non-discerning child saying, "If the child was non-discerning, his guardian should make the intention for Ihraam on his behalf. He should take off the form fitting clothes of the child and perform Talbiyah for him. By doing this, the child enters the state of Ihraam and all the prohibitions of Ihraam that apply to adults also apply to him. As for the baby girl that is under the age of discerning, her guardian should make the intention of assuming the state of Ihraam for her and make Talbiyah for her. By doing this, the girl enters the state of Ihraam and all the prohibitions of Ihraam that apply to adult women also apply to her. The children should be pure at the time of performing Tawaaf because it is like prayer and purification is one condition for its validity.”
There are two opinions regarding the purification of the non-discerning child during Tawaaf. The sounder opinion according to the Shaafi‘i School is that purification is a condition for the validity of Tawaaf. The jurist Ibn Hajar Al-Makki said, "Purification of the child and the guardian during Tawaaf is a condition for its validity. Thus, the guardian should perform ablution for the child and make intention on behalf of him."
Allaah Knows best.