Has the zakat al-fitr fallen from my brother, or was he required to have the intention of paying the extra amount, given that I paid his zakat at a higher price?

If your brother appointed you as an agent to buy three saa' (measures) of rice for Sadaqat al-Fitr and distribute it to the poor, and you bought it for an amount exceeding what he authorized you to spend, then the purchase is valid. However, the excess amount is binding upon you because you did not seek your brother's permission for that additional expenditure. Thus, your brother's obligation of zakat is fulfilled, and the excess amount is your responsibility, unless your brother voluntarily absolves you of it.