Zinc air will work fine. They have a really steady discharge curve, unlike an alkaline battery. The alkaline battery has a bit more voltage to begin with. But at the halfway point in these two batteries (alkaline and zinc air) lifecycles, they should be running around the same voltage (around 1.2-1.3v). On the second half of their life cycles, the zinc air will have more voltage. So in reality, they should operate very similarly over the course of their lifespan.
Also, the K1000 meter runs on a modified Wheatstone bridge circuit, like the Spotmatic F. Therefore, the meter circuit is very tolerant of higher and lower voltages, so the voltage change shouldn't effect the meter's accuracy. So this particular camera shouldn't have any issues.