How to switch DMM between 10 different batteries?
There are all kinds of solutions to automate this. Two spring to mind.

If your accuracy is simply .1V then you could use 10 resister voltage dividers going from the top of each battery to ground and multiplex that into a micro, measure the voltage sequentially on the top of each battery and do the math there and show the results on some suitable display.

You could use linear optical isolator across each battery to translate the voltage on each down to the right level to feed via a multiplexer or slide switch into a meter or again into a micro/display combination.

Or if you only care about battery volts &lt;10V or some such, maybe just a led indicating that for each battery. Maybe even 10 bar-graphs of leds.... 



However, since you mention trays of batteries, and unless you want to collect the data in a computer or something, the cheapest (counting your time), quickest, and least effort solution would probably be to just buy 10 cheap volt meters.