I agree its always best to get advice from a lawyer to be clear on your position, and we are not lawyers. That said, I have a client who provides SAT training services like you describe. I am not familiar with their legal stance or any requirements, but I do know that they address the following on their site.
SAT (and other tests for that matter) are registered trademarks of specific entities. With the SAT it is the College Board. Their web site clearly states this (as the last thing on every page), and that the related entity (SAT/College Board) was not involved in the production, and they do not endorse, the product.
Can you research other websites/businesses further and try to identify what others are doing and what they are addressing? That would be my plan. If there is a government process, there has to be open/public records of it. You could also do some direct research yourself first. Call a school contact you have if necessary.
Part of the consideration is, I believe local laws may play a role and those can vary. But it is possible to do what you say, as there are many who do it. If it is a barrier to entry, that may serve as an advantage to reduce competition once you have established yourself. For example, my daughters schools are members of the Florida Association of Independent Schools & National Association of Independent Schools. Can you get endorsements from specific organizations (like those mentioned) that can reinforce your products credibility? Best of luck along the way!!