DBA a 501c3 charitable organization without authorization to do so;
actually a 501c3 with IRS documentation to prove it;
or is entirely beholden to your ISD.
There's no ifs, ands, or buts there. Legally, they MUST be one of those three to operate.
Therefore, show up to a meeting or ask the president for a copy of the bylaws. They are legally required to provide them if they are the first or second example - they're charitable organizations or a charitable trust.
If they don't have bylaws (or won't provide a copy) then go over their head to your school district counsel, with your superintendent's permission. Tell them the story. Either they're doing business representing the school district in an illegal or unethical way (which will get them shut down) or they're flat-out illegal. Your district will jump on that really, really quickly.
If all else fails, call the IRS and ask them to audit the booster club. Poof. Gone.