What is the ruling on paying a fee to a cousin in exchange for him withdrawing a sum of money from the bank and converting it to cash, in order to bypass the daily withdrawal limits?

There is no harm in transferring money to your cousin's account for him to withdraw and give to you in exchange for a fee, as the money is yours and this is a permissible act. However, if he gives you the money before the transfer, it is a loan, and it is not permissible for him to take an increase on it, because that would be usury (riba). Ibn Qudamah narrated a consensus on the prohibition of every loan that draws a benefit. If he is to give you the money before the transfer, then it should be done gratuitously.