What is the ruling on one partner stipulating to the other that he alone manages the project? And what if the partner exclusively managing the project is an Islamic bank? And does expertise allow one of the partners to manage alone?

It is permissible to stipulate that one partner exclusively undertakes the work without the other. The working partner's share of the profit must be greater than their capital contribution, because their work deserves remuneration, as they combine the elements of both Mudarabah (profit-sharing) and 'Inan (equal partnership).