Unfortunately the physics of bat hitting ball is far more complicated than has been touched on and the actual maths behind it gets quite complicated. You cannot simply take two bats hit some balls and then, because you can't discern a difference, conclude that properties of the wood are negligible. If we were to try and develop a bat performance index there would be a huge number of calculations. Which is probably why no-one has done it. The key factors will be;
The moment of inertia of the bat; this is the torque required to alter the angular velocity of the bat, it's more relevant than dead weight,
Node locations and center of percussion; This is essentially the sweet spot, the area where maximum energy is imparted to the ball due to less energy being dissipated as vibrations. The bigger the sweet spot the better.
The stiffness of the bat face; Something to quantify the rebound characteristics of the bat, We could use the Coefficient of restitution however I'm pretty sure that would be relative to velocity. I imagine low density willow would allow the bat to have greater structural rigidity, carbon backing/inserts would have done a similar thing before they were banned.
This would allow us to create an index which quantifies performance by relating the force required to swing the bat to the resultant velocity of the ball. We could determine how often a junior, club player, semi pro and professional player would hit the sweet spot and then assign weightings accordingly. Equating the three would give us a figure quantifying performance.