I agree with @G07b10 - schools don’t want to negotiate. It is embarrassing for the family that cannot afford more to be asked to dig deeper. And the family who can pay more but doesn’t want to is probably not going to be a great part of the community.
But schools realize that occasionally a relative might come up with some help, for example, so you wouldn’t be the first to change your FA application after decisions. Totally agree that it’s better to explore this before applying.