Is it permissible, Islamically, to enter into a Mudarabah contract with an Islamic bank by converting a sum in USD to the local currency, with the bank guaranteeing the return of the original capital in USD in the event of a change in the exchange rate, so that the client recovers their original capital in USD without profit or loss?

A condition for the permissibility of a Mudarabah (definition) contract is that the capital is not guaranteed. The Mudarib (agent) is only liable if he exceeds his authority or is negligent. The correct condition is that the capital provider bears the loss, not the Mudarib. It is permissible for the bank (the Mudarib) to voluntarily bear the loss if it occurs, and this would be considered a gift to the client. However, it is not permissible to stipulate this from the outset, even as a promise. Therefore, it is not permissible for the bank to commit to compensating you if the capital and profit do not reach an amount equivalent to the capital in dollars, because this falls under guaranteeing the capital, which is religiously prohibited.