Putting all those instruments through a PA and not using a mixer is impossible. I assume you meant that not all instruments will be fed through the mixer and P.A., ie., bass, drums and guitar. My suggestion is to look into what type of mixer they are using. Often you can configure the mixer the way you want to and use one of the monitor mixes for the PA, while using the mains for the recording mix. If this is so, then I would suggest that you mic all instruments, place a stereo pair of ambeint mics in the field, pan the instrument to their natural place on stage and mix a live recording. This will give you the best possible recording.
If they are using a cheap headamp mixer and there is no flexibility then do as suggested above. But just be aware that the vocals will sounds as if they are coming through some cheap ass speakers.