Hi,i cannot see a reason why you would not have the original 6V battery.
You will struggle to find an off-the-shelf dropper,to cope with heavy 6V use,from a 12Vbattery,of which the starter will be primary.
You could convert accessories,starter and charging system,to 12 volt,but this would be expensive and may de-value a collectors marque.
You could use a step-up charge controller,to charge a 12V battery from a 6V system,but you would have two batteries,and have to keep that separate,although the earthing would be common.
I am certainly not advocating this,but......as a lad,i had a bench DC power supply,with 6 different voltages,obtained by having an old battery from me mum's Triumph Herald,with 4 carefully fitted woodscrews,allowing 2.2 volts to 12 volts,in steps of 2 ish volts
