Mobile edge computing (MEC) is a promising paradigm to handle the mismatch between computation-intensive applications and resource-limited devices. Nevertheless, as most Internet of Things (IoT) terminals are battery-limited, the computation gain of MEC may be compromised due to insufficient battery energy for task offloading. Wireless power transfer (WPT) and full duplex (FD) communications are economical charging and transmission methods for battery-limited IoT terminals. However, when integrating wireless power transfer and FD into MEC, the incentive problem should be jointly addressed with task offloading, because the WPT facilities and their powered IoT nodes belong to different service operators. In this paper, we investigate the efficiency of WPT from the perspective of battery saving, and propose an efficient wireless powered task offloading and incentive mechanism in FD MEC-enabled cellular IoT networks. The battery saving efficiency, which addresses both the total cost of WPT and saved energy of battery, is proposed as the performance metric. By adopting this metric as the utility function of the network operator (NO), the task offloading and incentive problem are jointly formulated as a Stackelberg game. We then propose an efficient alternating direction iteration-based algorithm to solve its equilibrium efficiently. Simulation results demonstrate the benefits of our algorithm in battery saving by comparisons with utility oriented benchmarks. Moreover, it reveals the tradeoff between the utility of NO and battery saving, which verifies the positive effects of FD communications and WPT in improving the efficiency of battery saving.