Can't pull more than 1.5A continuous, the lantern battery would run down in less than one day!
Kend - you got the right idea, solder the part on the lead wires like that. The regulator has 3 terminals, Vin, Vout and Com. Vin is (+) from the 12V battery. Vout is (+) to the electric fence. Com is (-) for both battery and fence.
The capacitor goes across (+) and (-) on the 6V for the fence. It's necessary because the fence delivers pulses and we have to make the regulator look like that battery. The capacitor has a low impedance that will do the job.
As to which regulator is the most efficient, generally speaking a switching regulator is the most efficient because it works like a transformer. Everything else just "burns off" the remaining voltage - HALF of it in this case.
That said, crutchow's point about the quiescent current is an important one, but I don't think it will be significant. You would have to have a pretty big idle draw from the regulator and a very low average draw from the fence to make it worse than an LM317T or a bunch of silicon diodes.