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This paper studies the joint beamforming design and time allocation for a two-way energy harvesting (EH) relay network, where a single-antenna base station and a single-antenna user communicate with each other through multiple-single antenna EH relay nodes, and an external multiantenna power beacon wirelessly power the relays. Unlike the existing studies that assumed the symmetric two-way relaying scenario, we assume a more practical scenario of asymmetric two-way relaying considering different traffic demands on the downlink and uplink. In this paper, we design the optimal beamforming vectors at the relays and at the power beacon jointly with the time allocation between information and EH transmissions, in the sense of maximizing the sum rate of the asymmetric information transmission under the EH constraints at the relays, and the power constraint at the power beacon. The formulated optimization problem is nonconvex, and thus, it is difficult to solve. To solve this challenging problem, we propose an iterative algorithm based on the semidefinite relaxation and the successive convex approximation techniques. Also, to reduce the computational complexity, we propose an efficient noniterative scheme based on the maximum-ratio transmission. The performance of the proposed schemes is demonstrated through simulations. [ABSTRACT FROM AUTHOR]