Nikon SB-600s Don't Work with 2700mAh Batteries?

Mah is equivalent to fuel in a battery, but Voltage is what runs the thing.  Without enough voltage you would get the response you describe.  However, the fact that they work fine in another flash is pretty telling, and I suspect your issue is physical.  I have seen these rechargeable batteries sometimes have less of a 'nub' protruding on the positive side of the battery.  If the flash's battery compartment is strict on size, that may be the case.

Though in no way the same, I recently bought a wifi card for my camera, but it would not work in my new 'semi-pro' camera.  Every other card reader on a camera took it except for that one.  In the new camera the card had to be pried loose when ejecting it.  Though there was n obvious size difference, the eye-fi guys replaced the card, and the 'identical' replacement works just fine in the new camera.  My point is that in this case despite what you see, the issue is probably just a minor physical difference causing the trouble.