An apparatus includes a bootstrap accumulator having multiple fluid expansion volumes each configured to receive fluid. The bootstrap accumulator also includes a piston assembly configured to move within the fluid expansion volumes based on pressures within the fluid expansion volumes. The piston assembly includes (i) a fluid pathway that couples the fluid expansion volumes and (ii) a bypass valve configured to selectively open or block the fluid pathway. The piston assembly could also include multiple pistons and a connecting rod coupling the pistons. The fluid pathway could include a narrower path through a first portion of the connecting rod and a wider path through a second portion of the connecting rod. The bypass valve could include a ball and a spring configured to push the ball to block the narrower path through the first portion of the connecting rod.