Would it be considered a cause for a secretary to lose her livelihood if I were to take her place in my brother's office, given that he will fire her to hire me?

There is no objection to working in your brother's office, even if it entails dispensing with the secretary, provided that her employment contract has ended or allows for her dismissal. It is worth noting that a woman's work as a secretary is often surrounded by Sharia-related caveats, and whenever these are found, it is necessary to stop them or to stop the work.