I assume your child already has a passport. In that case, then, simply apply for an ESTA on their behalf.

If ever questioned about it, you are a third party (e.g. parent) applying for an ESTA on behalf of your child. There is no problem with this. It's more common that parents think their non-ticketed infant does not require a passport or ESTA, when in fact they need both. So it's the right thing to get the ESTA for them.

CBP can and does care about minor children traveling without parents (e.g. trafficking or kidnapping) , but if both custodial parents accompany the child to the border it is not usually a problem at all. No one will care that the ESTA application is only for the minor.