With the engine running, the voltage at the battery should be 13.5v (charging system ok), so it does sound like the alternator or regulator is at fault. Here's where you need a service manual, as it tells you how to check each element. Nobody has weighed in with whether the ZZR600/1100 electrical systems are similar, and I'm afraid I don't know if they are, so I hesitate to say "Do this...." in case they aren't, and you end up blowing something! All I can suggest is you check all the wiring /connections between battery/alternator ( disconnect battery first), remove the alternator cover and inspect everything you can see for corrosion,etc. Pull apart any connecters and inspect for corrosion on the pins,dirt,moisture,etc. Use your multimeter to measure the wires ( lowest Ohms scale ), a length of wire should be 0 (naught) ohms, in other words a dead short or the same reading you get when you touch the 2 meter leads together. Sorry if you already know this.
Her's a link to a US ZZR600 manual, only one I could find. Warning: it's about 85mb but it's free.
http://www.shsu.edu/~rjm002/pdfs/ZZR600SvcMan.pdf