I've powered the Mini-Breakout board through the 7-15V input using 12V. Notice that there are two inputs to power the Mini-Breakout board with 7-15VDC: the J21 header and the J17-P4 pin. Although your battery is closed to 15V when it is fully charged, I don't recommend you to power the Edison with it. There is a chance that it might damage the board. I'd rather use another power source with an output voltage between the safety limits, for example 12V.
However, another option could be using a diode or two in series to the battery. The diodes will reduce the voltage to keep it below the 15V limit. For example, a silicon diode has a forward voltage usually of 0.7V. If you use two diodes in series to the battery, the two diodes will require 1.4V so you will get a voltage of 14.1V when the battery is fully charged. I haven't tested this way, and as I said above, I would rather use another power source, but you could give it a try.
A third option would be using a zener diode to regulate the output voltage. Here you will find a brief description about how to use zener diodes as a voltage regulator: Basics: Introduction to Zener Diodes | Evil Mad Scientist Laboratories
Yeah, I agree with you, it might damage the board.
Thank you for the suggestions!