Maintaining precise pressure regulation in hyperbaric surgical chambers is critical for patient safety, particularly during dynamic disturbances such as door operations or environmental fluctuations. Conventional control strategies often fail to balance rapid response and stability under such conditions. This study compares two advanced methodologies: an adaptive Proportional-Integral-Derivative (PID) controller with neural network-driven parameter adaptation and a model-based feedforward controller, for hyperbaric chamber pressure regulation. The adaptive PID employs real-time tuning via a model reference adaptive system (MRAS) to dynamically adjust to disturbances, while the feedforward controller leverages a first-principles pneumatic model to preemptively compensate for anticipated disruptions. Both strategies were validated through simulation and experimental trials under realistic scenarios, including abrupt pressure changes and thermal variations. Key metrics - settling time, overshoot, and steady-state error - were quantified. Results demonstrate the adaptive PID reduces settling time by 32% compared to the feedforward method and achieves superior disturbance rejection. Conversely, the feedforward controller exhibits better steady-state precision under predictable conditions but suffers vulnerability to model uncertainties. A hybrid architecture integrating both strategies is proposed to synergize anticipatory and reactive control. This work provides practical guidelines for designing robust pressure regulation systems in safety-critical medical environments.