@siv, a couple of thoughts… if you don’t feel like your current school is a great fit for him, getting a scholarship isn’t going to make it a better fit. It might save you money if you don’t have other options, so if this is your concern, you may want to apply. And note when you would have to accept. (Day schools are often on an earlier cycle than BS, which is its own mess!)
It doesn’t sound like you are in a situation where you are deciding between a full scholarship and full pay. If you need $10,000 a year to make it work, apply for that. Most schools strive to have kids across the economic spectrum. It creates an awkward community when everyone is either well enough off to write that check or can’t afford it at all.
If you need $10,000 so that you don’t have to give up the country club membership, well that’s a different story. Donors aren’t funding scholarships for that reason. But they do want the school filled with kids who are going to make the most of the experience and realize it’s not all or nothing.
The fact that you can contribute something meaningful and are willing to do so matters. If you are really confused about this, you can raise it with the AO.