Scot Graham, 55, who was offered a $200,000 settlement last month, is now upping his demands, saying his identity shouldn't have been made public.
His attorneys say the LAUSD's public identification of Graham invaded his privacy and placed him in a false light. They're now seeking more money for Graham.
District officials voted Tuesday to unanimously reject the counter settlement proposal from Graham's lawyers.
Cortines, 79, said the activity between him and Graham may have involved bad judgment on his part but the encounter was consensual.
Under the earlier settlement, Graham also would have received lifetime health benefits, estimated to be worth $250,000, for leaving his $150,000-a-year job as the director of leasing and facilities at LAUSD.
The district has not commented.