My question is whether it is possible to return to the playa on Monday afternoon.
Yes, but you would need your ticket stub and in-out wristband that you acquired on your way out.
Obviously it will be impossible to enter by the 9-mile gate.
Actually, that is where you would return. You would be directed by Exodus staff accordingly and go outside the flagged lines to be processed by the gate.
But can I arrange to enter the playa by the 12-mile gate?
Highly doubtful and you would probably be sent back to the main gate anyway.
Who might I contact to arrange this BEFORE the event?
Sorry, but 8/9 mile is your best bet, though I don't fully understand why you wouldn't just pack up on Sat. and be done with it.
ETA - looks like Stretch beat me too it. That's what I get for being ADD and leaving the window open for a while.