A LOT depends on the school. And some depends on how badly they want your child to be a student there.
Instead of pulling out and going elsewhere (which will not do you any good with School 1) contact school number 1 and say that it is your child's first choice school. See if there is any possibility of increasing his aid so he can attend.
I will say...if you are asking for financial aid money to cover your FAMILY contribution...you are unlikely to get it. If you are asking for additional money to cover a gap in financial aid, it all depends on the school.
When our DD got her aid packages, she talked to two schools. One school FLATLY refused to even discuss their financial aid awards...wouldn't even talk about it. The second school gave her an additional student loan and added work study to her package. They did not increase her "free money" grant/scholarship.