To improve the heat dissipation performance of the battery module, the fins are embedded in a hybrid battery thermal management system (BTMS) with phase change material (PCM) and air cooling. To balance cooling performance and energy consumption of BTMS, the variable-section fins inspired by the features of the beak and wing are proposed. Subsequently, the effects of the thickness of PCM, fin type, airflow velocity, and support frame width (L1) on cooling performance and power consumption of the battery module are investigated using the computational fluid dynamics (CFD) model at high rates of 3C charge and 4C discharge. The maximum temperature (Tmax) of BTMS adding fin is significantly decreased by 3.4 °C at an inlet velocity of 3 m/s. Moreover, the power consumptions of BTMS with beak fin are reduced by 23% and 33% respectively compared with rectangle fin and wing fin. Additionally, the influences of structure parameters of bifurcated fin, such as angle (α), the length of end exposing air domain (L2), and windward shape on cooling performance, temperature difference (ΔT), and power consumption are comprehensively analyzed. The results demonstrate that under the bifurcated fin α of 75° and L2 of 40 mm, the Tmax, ΔT, and power consumption are 38.3 °C, 2.9 °C, and 2.6×10-3W respectively, a decrease of 0.4 °C, 0.4 °C, and 13% compared with BTMS with beak fin. Furthermore, the cooling performance of BTMS introducing delayed air cooling is researched, and the power consumption is observably reduced by 59%.