The batteries of the same voltage will put out power for a longer time with more milli amps, not more voltage. They will just last longer before being recharged.
Radio control uses lipo batteries for their power in a lot of their motors. A 1 cell lipo puts out 3.7 volts. A 2 cell puts out 7.2. As you go to higher cell count, add 3.y volts for each cell. The milli amp designation is how much power the battery can hold. So, if your lights are brighter, then the voltage is higher, not the amount of power the batteries can supply in milli amps.
Your lights may last for a while with the increased voltage, but I would be cautious on applying more voltage than designated. This would be like running a 6 volt bulb on 12 volts. won't last too long.