[HayZee518]

2/3 copper is perfectly legal and don't worry about the derating factors. Being it's such a short distance away it doesn't even enter into the calculation. The four wires are part of a cable assembly so derating isn't necessary.
In your sub panel don't tie the neutral to the enclosure ground with the screw or jumper. Install an auxillary ground bar and tie the bare copper from the 2/3 cable to this bus. Scrape the paint away from where the two screws connect the bar to the case. The red and black go to the main lugs in your sub and the white goes to the neutral strip. For branch circuits all bare grounds go to the aux bar, all whites to the neutral.