No, it is not a good law. Previous law and Courts have ruled that if government requires something of a citizens (in this case, education), then the government must meet the same Constitutional requirements afforded citizens in the Constitution when meeting that requirement. Thus, if Government requires you educate your kids then they have to uphold the same Constitutional mandates within that requirement. Thus, you cannot require me to send my kids to a public school and at the same time require that if I send my kids to school that I surrender other rights afforded by citizens. Thus, testifying or proving my citizenship would be an undo requirement because it would force citizens who have no reasonable suspicion to prove their status, thus a violation of the 4th for all citizens.
However, in the case of Drivers Licensing (often used against my viewpoint), it is not required to have a license therefore more requirements can be placed on obtaining a Driver's License than upon getting a public education.
The argument in education is that there are other options. However, the default option is public school. Thus, if you don't educate, you could be forced to send your children to the public school or face penalties. As a result, you cannot be asked to appease inquiries into possible illegalities that fail to meet the reasonable suspicion clause when fulfilling this requirement.
While I do not like the rule requiring mandated education, it is right that they cannot ask for more documentation on something that government mandates.
Thus, this is a violation of the 4th.
Now, you said that the ruling violated the Constitution, you never explained how it violated the Constitution.