You will find many instances in which you have to provide a social security number for your children. Claiming them on your taxes is one that comes to mind.
Many places now just need to see the social security card to verify that the number you've provided matches. We require proof of the right to work in the US, but we don't keep actual copies of it.
I had a business as a client who had an employee who provided a social security card. After being there for the probationary period, they submitted the paperwork to have him added to the group health insurance. Lo and behold, the insurance carrier caught the fact that the number actually belonged to one of their former subscribers. Former...as in deceased.
I understand you wanting to protect yourself, but unfortunately, people lie. People commit fraud. If you want you children to have the benefits of an education at a school that receives government funding, I don't see any way around you providing the social security numbers and allowing the school to at least verify them.
Talk to the school about your concerns. Hopefully, they can allay your fears.
Best wishes.