As in tennis, you can consider there being two sweet spots on a cricket bat. One is a performance, or nodal, sweet spot where the sum of the amplitudes of the three dominant vibration modes is a minimum i.e. absorption of impact energy is minimised. The second is a comfort sweet spot known as the Centre of Percussion, where the forces of rotation and translation cancel out (excluding rotations for off-centre hits). The COP is rarely in the same place as the nodal sweet spot, and usually slightly above. By this reasoning you will get the best batted ball speed from a shot that you will 'feel' a reaction in the hands, and a slightly reduced batted ball speed from a shot out of the COP, which most of call 'middling it'.
Heavy bats absorb less impact energy than light bats, and will likely have a larger nodal sweet spot, but mass distribution will effect size and position.
I have a paper which compares various profiles' performance against impact vibration modes, which I can email to those who are interested.
Talisman, I will respond to some of your questions shortly.