AFAIK, metal boxes need their own ground, independent of the grounding on the outlet. Typically done with a pigtail that is screwed to the box and than wire nutted with the ground wire coming into the box. You need to check what size is needed, probably much smaller than your 6AWG primary cables, I'd guess 12 or maybe 10 is probably what you need. Premade pigtails with the grounding screw attached are handy when you're doing a lot of boxes.
There are automatic devices meant for car chargers that enable one of two devices to be used, both for sharing between two cars and for sharing an outlet between a dryer and car, for example. A Neocharge smart splitter is an example (I have no experience with these). Something akin to a generator transfer switch might also work, ask your electrician.