Electrically Grounding a Large Heatsink to PCB Layer Ground
Is there any reason I should NOT be connecting the heatsink screw hole
  mount pads to this middle ground layer?


One reason not to is that you may turn the heatsink into an antenna and unintentional radiator if your ground is especially noisy.  

This is an unlikely scenario though and is very much dependent on the design. In most cases grounding the heatsink electrically will help cut back on noise. 

The other reason is if the heatsink could bleed heat back into the PCB in an area you may not want it.

Something to think about and design for.