Is it permissible for my mother and me to perform 'Umrah through a usurious bank where my brother works, given that the bank reduces the cost for employees' families, and knowing that we pay the full amount ourselves?

You must advise your brother who works in a usurious bank and explain the gravity of assisting in usury. As for performing Umrah through the bank, if the bank does not pay any of the expenses, then it is permissible for you to go. But if it does pay, then it is disliked for you to benefit from what it pays, due to the mixing of the bank's money with unlawful (haram) funds. Seeking lawful money for Hajj or Umrah is one of the most important reasons for acceptance.