Single Ground Wire Run Under Floor Joists
Your #2 wire is large enough to be run without protection or a backer if running perpendicular to the joists in the basement ceiling. 250.64.B.3 requires wires smaller than #6 to be protected. The basement ceiling is considered a safe location and the specific height is not given. 334.15 provides the examples of attachment as you have mentioned.
I would caution that some older code answers are not current code as code has had significant changes in the last few cycles. (Each cycle is normally 3 years).