You need to tell them (in the most diplomatic manner possible) that they can't have their cake and eat it. The can have the work at a fixed price if they want,

You need to tell them (in the most diplomatic manner possible) that they can't have their cake and eat it. The can have the work at a fixed price if they want, or they can pay for time but it's unrealistic to expect both. It sounds to me like they were very unhappy with the job that took longer than estimated, and you need to address this if it's the root cause of their change.