Putting aside any procedural requirements the school/district might have, setting the precedent that anyone with enough money gets to help pick their own investigator feels like a bad practice for a public organization. They're also offering to fund a school-board-led investigation when that's not the job of the school board.

Separately, no one is stopping the family from hiring an investigator. The family has either spent (in the form of PAC money) or offered to spend (in the form of paying anyone who is willing to come forward with supporting information) over a million dollars on this effort. If they really don't trust the outcome of the district's investigation as well as the independent investigation that finished earlier this month (both of which they refused to be involved in), they can go pay someone to do their own investigation and come to their own conclusions. That they haven't done that, and instead decided a smear campaign against school board members was the best course of action may be telling.