One thing you may try, the board of ed probably has rules, if it's over a set dollar amount it has to go out for bid. Under that amount, say $600 to $700, they may be able to write the expense as small incidental ongoing repairs and just hand out the work with no bid.
Your contact person, you may offer to work T&M and split the invoice into two or three pieces to stay under the mandatory bid process rules. It can be time consuming and a hassle also for the school to shop around and run a bid process, documentation, for small jobs.
You can offer to save them (and yourself) time and money to just come there and get the work done instead of going there just to give the answer they require given, regardless of right or wrong. It can be a test of if they want to be friends or adversaries.
It was ~ 1995 and we changed the boilers at the local HS. I had done a lot of them, we were longtime low bidders, very competitive. Long story short, we had to win the bid twice over a 1.5 year period. The second time with our design correcting just huge, joke huge, design mistakes. The second bid was $100,000. less than the first one, included right sized boilers, a third boiler for a dedicated DHW system with multiple tank in a tank heat exchangers, and a $9000. Heat Timer electronic reset boiler control.
Time came to do the job and I knew the guys would not be able to see what they were doing (quality of work issue). I took down a few incandescent fixtures and put up three 8 ft HO flourescents. I wrote the change and planned to just give the lights away if they refused the charge. Dummy me, thought I could make it up in the future. The permanent building committee refused the $600. invoice and ordered me to take the lights out. I took them out after completion, put the old lights back up, and it became hard to see your shoes with the lights on again.
I did not care too much about getting screwed on the invoice but was incensed they would refuse my generosity. They were interfering with the quality of my work.
Probably ten years later I was down there voting and brought in a digital camera to take some job photos. The maintenance super remembered me well. They had put an addition on the building which I would have bid if I was not too busy. I saw they had added a third boiler for the guesstimated load, they were again clueless, saw it was not piped correctly with primary and secondary pumped loops. It was piped straight parallel so the idle boilers would have full hot water flow and full heat loss up the chimney.
Took a picture of my work then turned to snap a picture of the $9000 Heat Timers running the fully modulating fire burners. My jaw just snapped shut. There was a perfect impression on the wall where the boxes had been, clean rectangles on the dirty wall. They had taken the burner controls out when they installed the BAS. They did not know I could have made good money installing those boxes in another school, not to mention that the BAS was probably a lot worse at the specific task of fully modulating burner fire control with hot water reset. Fuel savings from the Heat Timers could be over $25,000 annually.
They probably never saw anyone move so fast from the boiler room, through the voting booth, and out the door. Probably the last time I voted in that town or drove down Main Street.