The AVIC switches between day and night view based only on the time of day and your location. The illumination wire doesn't affect it at all.
The illumination wire switches the brightness level of the display. You get only two brightness levels (both selectable in one of the menus). Ideally, you would connect the illumination wire to one of the parking light feeds since that would work correctly no matter what the instrument dimer switch was set to. However, the LS makes that very difficult. Instead, connect it to one of the dash light illumination feeds. The easiest two to get to would be either the bulb for the ash tray illumination, or the illumination wire going to the message center buttons. Connecting to either of these will work as long as you don't turn the dimer to a very low setting.