The high voltage is of no concern to your battery but you are right that you are not going to get best output. From Ed's description they are high voltage devices and for best results you would be advised to rewind as he suggests. Ideally with the minimum turns to give you the cut in speed that you need. That way you could trade volts for current.
You could also use a dc/dc converter or a transformer and rectifier, either of these methods will give additional losses that may cost you more than you gain from the tiny machine.
I would suggest that you measure how many volts you get at your cut in speed then reduce the turns and increase wire size to give you 12v cut in at that speed. Just change turns in ratio of original o/c volts to 12v. That way you will get the best possible output. if you use it as it comes then I think you are looking at about 25mA which is hardly enough to supply the battery self discharge.
Flux