Is it probable that a 15VDC device be damaged by a 14-16VDC battery?
The voltage input tolerance is specified in the manual for the Thorlabs KBD101, pp 63. See screengrab below. 

It clearly states the voltage tolerance to be +/- 3% - 14.5 V to 15.5 V. It's impossible for anyone of us to tell you any different without knowing what goes on inside the unit. It might handle a wider tolerance, but I'd not recommend you trying, that's an expensive bit of gear ($731 at time of answer) to experiment with.