Does one break their fast if they feel food and acidity rising to their throat without expelling it, and must they make up the days on which this occurred, especially since she insists on making them up despite fatwas stating it is not obligatory?

If one feels food rising and tastes it in their throat, their fast is not broken unless it exits to the mouth and is then swallowed intentionally. If it returns to the stomach unintentionally, there is no compensatory fast required, because the mouth is considered as an external area.