This study presents BIRDSHOT, an integrated Bayesian materials discovery framework designed to efficiently explore complex compositional spaces while optimizing multiple material properties. We applied this framework to the CoCrFeNiVAl FCC high entropy alloy (HEA) system, targeting three key performance objectives: ultimate tensile strength/yield strength ratio, hardness, and strain rate sensitivity. The experimental campaign employed an integrated cyber-physical approach that combined vacuum arc melting (VAM) for alloy synthesis with advanced mechanical testing, including tensile and high-strain-rate nanoindentation testing. By incorporating batch Bayesian optimization schemes that allowed the parallel exploration of the alloy space, we completed five iterative design-make-test-learn loops, identifying a non-trivial three-objective Pareto set in a high-dimensional alloy space. Notably, this was achieved by exploring only 0.15% of the feasible design space, representing a significant acceleration in discovery rate relative to traditional methods. This work demonstrates the capability of BIRDSHOT to navigate complex, multi-objective optimization challenges and highlights its potential for broader application in accelerating materials discovery.