Electronic – Can we make a rechargeable zinc air battery at home

Tadiran Batteries have LiSOCl with an integrated super capacitors. Those are not rechargeable and are quite expansive compared to the more common CR types. The standard LiSOCl AA cell can manage 100mA/100mSec pulse at ~3V (Dropping from 3.6V). There are many manufacturers of such batteries (Tadiran, Saft, Xeno, EEMB and others). Of course, there are also D sized cells with 19Ah capacity and much higher pulse capabilities.

Edit: see also Is soldering wires directly on a NiMh battery safe?

nothing in there that I could see that would 'balance' them

This may be why they don't last very long.

Due to manufacturing variance, one of the batteries will go flat first while the others have more charge. This can result in current being driven through it by the others while it's flat, damaging the battery. Or the reverse happens while charging: one will fill up first, and then suffer overcharge damage. Does the charger automatically stop on full charge or it it timed?

It may be better to adapt the battery holder to the batteries with a saw, so you can use the holder and charge them individually.

Or look for cheap repurposeable packs, eg. http://www.dx.com/p/gd-509-2-4v-1200mah-rechargeable-2-x-aa-cordless-phone-replacement-battery-pack-white-292547#.VIXLYsmfjjU is clearly 2x AA heatshrunk together, you may be able to find 4 heatshrunk together in the right format then just change the connector.

Edit 2: by the way, I would put money on the tanks suffering mechanical failure or enemy action before the battery packs become an issue. Unless they're expensive 'models' rather than 'toys'.