Is it permissible for me to take out a loan in my name for my brother from a bank that finances with shares using the Tawarruq method, and then my brother repays the installments to the bank?

If you agree with your brother to conduct a financing transaction in your name to deliver the amount to him so he can pay a price, then this is permissible, and you are considered an agent for him, as he has appointed you to purchase shares in your name from the bank and sell them on his behalf. It is not necessary to disclose the name of the principal in the agency; rather, it is valid for the agent to purchase in his own name.

However, if you did not agree with him and made the decision yourself, then this is considered a loan from you to him, and he is only obligated to repay the amount you paid him. He has no relation to the bank's profits. But if he voluntarily gives you an increase over his debt upon repayment, there is no harm in that.