In this work, we systematically examined the structural stability, mechanical properties, and thermodynamic behavior of B2-type CuBe alloy, and compared the results with isostructural Cu-based alloys (CuAl and CuZn) by employing first-principles calculations in the pressure range of -22 to 100 GPa. This study revealed the stable existence of CuBe alloy at low-density expansion states (e.g. ~ -20 GPa), indicating its superior structural stability compared to CuAl and CuZn. The pressure dependence of properties such as a (a/a0) and r (r/r0), elastic parameters (Cij, B, G, and E), deduced parameters (B/G ratio, ν, Vickers hardness, sound velocity, and ΘD), and thermodynamic parameters (F, S, and Cv) were investigated. All CuM (M=Be, Al, and Zn) alloys had more difficulty undergoing uniaxial stress than shear stress. External pressure reduced the ductility of the CuBe alloy, while excess pressure (P>50 GPa) resulted in increased ductility, which was similar to CuAl but different from CuZn. The hardness and ΘD values demonstrated consistent variation corresponding to the ductility changes. Thermodynamic parameters were minimally affected by pressure, and the stronger interactions led to greater F in the CuBe alloy. These findings offer confidence for the future design of ordered Cu-Be alloys with exceptional properties.