I'm no trans expert, but I seriously doubt the contention that you can't let the trans run dry for a few seconds. The fluid is there primarily for hydraulic reasons. It also provides a cooling function (that's why it goes through a cooler).
There are no cams, pistons, cranks, etc that require constant and perfect lubrication like an engine has. The gears are big and robust. If the trans required perfect lubrication, we would change the transmission fluid every 3-5000 miles just like an engine.
Run it until it starts to sputter, then shut if off. It's in park while you do this, so how many trans parts can be turning anyway?
I've used this method other vehicles (not an LS) and it works great.
Dale
Dale