Im after a 12v hairdryer also with the colder months comming but i have never seen one for sale. (Not that I have looked before either).
Mike you describe your battery as an A200. That doesnt tell us the capacity of the battery. Something that generates a lot of heat will take a huge amount of power. You need to know the wattage of the hairdryer.
Lets assume the hairdryer uses 100 watts. A 12v 20 amp hour battery can theoretically supply 240 watt hours(12x20) total, or run that dryer for 240/50 = 2.4 hours. It wont be that good as amp/hour ratings are given for a 20 hour discharge.
I have a small 2 amp/hr battery only for running my dsc and that wouldnt get the hair dryer warm before flattening the battery.
Maybe a worthwhile project is a box with a lid for holding eyepieces, with a small fan and heat source to keep the eyepieces toasty on those cold winter nights