All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.

"Open buffet" refers to a designated place that offers food and drinks in exchange for a specific amount of money paid by the customer, granting them permission to eat from all the items displayed by the venue owner.

Eating from an "open buffet" is permissible in Sharia (Islamic Law), even if there is ignorance regarding the quantity of food consumed by different individuals. This ignorance is considered minor, and scholars may tolerate it in transactions. They have stated that a sale is valid despite minor ignorance. This is because the level of uncertainty is low and does not lead to disputes, especially when customs dictate such transactions. Both the provider of the food and the customer are tolerant of each other's rights in such situations.

Imam Nawawi, may Allah have mercy on him, said: "The general rule is that selling based on uncertainty is invalid ... What is meant here is that which has an apparent uncertainty that can be avoided. As for what necessity calls for and cannot be avoided, such as buying a pregnant animal with the possibility of the offspring being one or more, male or female, with complete or incomplete organs, or buying a sheep with milk in its udder, and similar cases; then this type of transaction is valid by scholarly consensus. The scholars have also unanimously agreed on the validity of transactions involving minor uncertainty, such as:

The entire Muslim community has agreed on the validity of selling a stuffed garment even if its stuffing is not visible. However, if only the stuffing is sold separately, it is not valid. They also unanimously agreed on the permissibility of renting a house or similar properties for a month, even though it may be for thirty days or twenty-nine days. They also agreed on the permissibility of paying a fee to enter a bathroom and the permissibility of drinking from water provided by a well with a small fee, despite variations in people's use of the water or their stay in the bathroom" [Al-Majmu' Sharh Al-Muhadhdhab, Vol. 9/P. 258] with slight adaptation. 

The essence of permission is allowing the payer to eat the food for himself from what is offered on the basis of permission, not ownership, as indicated by not allowing him to carry the food with him except with permission. Therefore, he does not have the right to give it as a gift or feed others, because he is not the owner of the food, and he must comply with the terms and requirements of the permission given to him.

The father of the child is not allowed to feed him from the food without the permission of the owner of the food, as long as the restaurant management stipulates that tolerance is only for children under the age of five. As for those older than that - as in the question - they are not permitted to eat except for payment.

In conclusion, it is incumbent upon the father of the child to contact the restaurant management, inform them of what happened, ask for forgiveness, and pay for what is required if the restaurant requests compensation for what the child consumed. It is known that anyone who damages someone else's property or consumes it without right must ensure reimbursement in kind or value. And Allah the Almighty knows best.