Normally, student passes are for international students who do not have any families here or are accompanied by a Pei du Momma (who is NOT allowed to work at all, at least for the 1st year. If a Pei du Momma is the situation then the husband cannot be here as that forms a family unit and is NOT what the student pass is designed for. If your family is here then it stands to reason that they are on a work pass of one form or another. In that case, the student will qualify with just a Dependent's Pass IF the parents qualify to have the child here in the first place.
No matter how you twist it, you are not going to get a student's pass for your child to study here in Singapore as long as you or your spouse doesn't make the required threshhold income as stipulated by the MOM to be able to bring in dependents. It's rough, but you should have found out all this stuff before you took up jobs in a foreign country that doesn't operate like your own country.