In this paper, a laser-powered aerial mobile edge computing (MEC) architecture is proposed, where a high-altitude platform (HAP) integrated with an MEC server transfers laser energy to charge aerial user equipments (AUEs) for offloading their computation tasks to the HAP. Particularly, we identify a new privacy vulnerability caused by the transmission of wireless power transfer (WPT) signaling in the presence of a malicious smart attacker (SA). To address this vulnerability, the interaction between the HAP and the SA in their allocation of tile grids as charging points to the AUEs in laser-enabled WPT is formulated as a Colonel Blotto game (CBG), which models the competition of two players for limited resources over multiple battlefields for a finite time horizon. Moreover, the utility function that each player receives over a battlefield is developed by identifying the tradeoff between privacy protection level and energy consumption of each AUE. We further obtain the mixed-strategy Nash equilibrium for the modified CBG with asymmetric players. Simulation results are presented to show the effectiveness of this game framework.