For a class of discrete linear time-varying cooperation-competition multi-agent systems, this paper studies the bipartite consensus control problem under energy-constrained denial-of-service (DoS) attacks. We consider signal transmissions from the controller to the actuator that are subject to energy-constrained DoS attacks deliberately launched by adversaries to block information communication. The concept of energy harvesting technology is introduced to describe the constraint of the energy required by the attacker to launch the attack, and a random attack model is constructed. The objective of this paper is to design an observer-based bipartite consensus controller for cooperation-competition multi-agent systems under energy-constrained DoS attacks, which ensures that a predetermined ${\cal{H}}_\infty$ bipartite consensus is achieved over a given finite horizon. By utilizing the Lyapunov stability theory, the sufficient conditions ensuring the desired ${\cal{H}}_\infty$ bipartite consensus are established, and the bipartite consensus controller scheme is devised through optimization techniques, and the controller and observer gains are obtained. Finally, a simulation example is presented to demonstrate the effectiveness of the proposed secure bipartite consensus controller scheme.