One question I would like to know, have you made it clear to the school that you will be leaving if the current offer is not changed?
I would have no problem with you announcing that you are leaving, and as you hold no contract I am sure it would be near impossible for them to force you to stay (more trouble than it is realistically worth). However I'm not sure they are aware what they are gambling if you choose to leave.
International schools are not like Thai schools, teachers for specialist subjects like Physics need a specific experience profile and are hard to recruit, particularly now when you need to pay ASQ to bring someone in and term time has already started. There is also significant backlash from parents who are paying 6 figure tuition fees and would expect specialist math and science teachers.
I'm sure if you made it clear to the school management that you would leave at this stage, without even needing to give them a notice period, they would realise exactly how difficult you are to replace. If they don't at least attempt to compromise, then I would suggest you make it happen. Seeing as you have made it clear are financially secure, it seems like it would be preferable to being short changed by these people.