In this paper, the performance of a  two-hop cooperative network is analyzed assuming the use of a channel-aware dynamic time-switching (TS) energy harvesting (EH) protocol at the battery-assisted EH full-duplex relay node. In such networks, node-level energy considerations are important for efficient battery utilization but have not attracted research attention. Two battery management schemes are considered. In the first scheme, fixed  battery energy (FBE) is added to the harvested energy.  To further improve performance, a new dynamic battery energy (DBE) scheme is considered in which the amount of battery energy drawn is channel dependent. The performance of the network is analyzed with both FBE and DBE schemes for static and dynamic TS schemes assuming a practical nonlinear EH model. Expressions are also presented for the average battery energy consumption in each case. It is demonstrated that joint optimization of  the average battery energy consumption and the TS parameter is important. It is also demonstrated that the proposed dynamic TS protocol provides substantial gains in throughput and average energy savings compared to the static TS protocol with both FBE and DBE.  The derived insights can aid a system designer. Computer simulations validate the derived expressions.