A ladder frame for the kick faced with the cabinet exterior material. Only solid wood touches the floor. So plywood on edge. No shims. Scribe to floor.
A power plane is your friend.
I wouldn't hesitate installing the floor over the whole floor. Standard practice. We refuse to use any "leg" to support cabinets. We build full kicks and scribe them to the floor. I would suggest anything you can do to distribute the weight over the islands footprint would be the best way to go.
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