Yeah, we were talking about a reserve in the initial BCG kinetic energy, which comes from initial BCG mass and velocity. Initial velocities are often nonzero, and I'm estimating 200 in/sec
Of course some of this kinetic energy goes into the spring as spring potential as the BCG moves rearward (see the equation in the OP), and then gets returned to the BCG as it moves forward.
I always thought that this ebb and flow of energy was one of the beauties of simple harmonic motion, which this is akin to.
Your "inertial energy" could only be interpreted as rest energy, and I suuure don't want to get relativistic in this system.

