Doubly-near-far problem in RF-powered networks can be mitigated by choosing appropriate device-to-device (D2D) communication mode and implementing energy-efficient information transfer (IT). In this work, we present a novel RF energy harvesting architecture where each transmitting-receiving user pair is allocated a disjoint channel for its communication which is fully powered by downlink energy transfer (ET) from hybrid access point (HAP). Considering that each user pair can select either D2D or cellular mode of communication, we propose an optimized transmission protocol controlled by the HAP that involves harvested energy-aware jointly optimal mode selection (MS) and time allocation (TA) for ET and IT to maximize the sum-throughput. Jointly global optimal solutions are derived by efficiently resolving the combinatorial issue with the help of optimal MS strategy for a given TA for ET. Closed-form expressions for the optimal TA in D2D and cellular modes are also derived to gain further analytical insights. Numerical results show that the joint optimal MS and TA, which significantly outperforms the benchmark schemes in terms of achievable RF-powered sum-throughput, is closely followed by the optimal TA scheme for D2D users. In fact, about $2/3$ fraction of the total user pairs prefer to follow the D2D mode for efficient RF-powered IT.