I use two RM640A batteries in mine. Unfortunately, after cleaning it up and fixing the contacts, the electronics are out on the shutter. Shutter opens, but does not close on its own. Lens is perfect, so maybe I can find another.
ANYWAY: The Zinc-Air batteries are suitable for low-current cameras that use CDS cells for metering. The HM-E draws a lot of power, and I'm thinking the Zinc Air batteries cannot deliver. Try two RM640A's.