The battery is just a 12-volt rechargeable (lead-acid) battery. The systems include one or two 150Watt inverters to convert that to 120VAC.
You could use any inverter with any big 12V lead-acid battery (including a regular boat or car battery - but better is a deep cycle battery), but you'd need to make sure that the inverter could meet the minimum current draw of your monolights. Actually a 150W inverter like they use is very small. Apparently the White Lightnings are willing to work with very low recycling currents (which other monolights might not be) and the inverters are protected against overcurrent draws. I would want a much bigger inverter, if possible - which would also lead to much faster recycling times.
By the way, those inverters look exactly like these:
http://www.batterystuff.com/inverters/