This work presents the design, modeling, and experimental evaluation of a biomimetic robotic fish actuated by shape memory alloy (SMA) wires integrated with a bistable polyethylene terephthalate glycol (PETG) caudal mechanism. Inspired by the general morphology and sub-carangiform-type swimming motion of Atlantic salmon, the robot employs a bistable elastic structure and a flexible caudal fin to convert SMA contraction into bidirectional tail motion through snap-through deformation. The kinematics, dynamics, and hydrodynamic characteristics of the robotic fish were investigated to evaluate the feasibility of mechanically amplifying the relatively slow SMA actuation into larger-amplitude tail motion. Water-tank experiments were conducted using a tethered prototype with a body length of 280 mm. Over 20 trials, the robot traveled 0.42 m with an average travel time of 39 s, corresponding to an average forward speed of 0.0108 m/s (0.038 BL/s), while the best observed trial required 31 s, corresponding to 0.0135 m/s (0.048 BL/s). The bistable mechanism produced approximately ±30° tail deflection, with a peak-to-peak oscillation frequency of approximately 1 Hz under the selected actuation condition. The measured acoustic output was approximately 45 dBA for the SMA-based system compared with 68 dBA for the motor-driven system, indicating a substantially lower measured acoustic output under the present experimental configuration. The swimming experiments also resulted in a high Strouhal number, primarily associated with the low forward speed and tether-induced drag. Overall, the results demonstrate the functional operation of bistable SMA-based actuation for compact robotic-fish propulsion and highlight its potential for reduced actuator-associated acoustic output. Further optimization of the SMA thermal response, bistable mechanism, tail kinematics, and untethered configuration is required to improve swimming performance and enable more complete evaluation of the propulsion system.