I leave my Leaf's Bluetooth OBD II scanner plugged in all the time. No issues unless you leave the app running. The Leaf has a half size or so 12V battery.
I have a 20A capable 110/220V EVSE. I just got my i3 REX tonight but I have owned a Chademo Leaf for about 14 months. At some point I'll compare/contrast on my blog. I have a Bluetooth OBD II, now to see what I can see. :)