This paper studies joint transceiver design for a full-duplex (FD) cloud radio access network with simultaneous wireless information and power transfer. In the considered network, a number of FD remote radio heads receive information from uplink users, while transmitting both information and energy to a set of half-duplex (HD) downlink users with power splitting receivers. We aim to minimize the total power consumption with both uplink-downlink quality of service constraints and energy harvesting constraints. The resulting problem is challenging, because various design parameters, such as the transceiver beamformers, the uplink transmit power, and the receive power splitting ratios, are tightly coupled in the constraints. Four different solution approaches are proposed for the joint transceiver design problem, each one leading to a different numerical algorithm. In particular, a block coordinate descent method is proposed, and by exploiting the problem structure, we prove that the algorithm converges to a Karush-Kuhn-Tucker solution, despite the coupling of various design variables in the constraints. Simulation results validate the effectiveness of the proposed algorithms as compared with the traditional HD scheme.