I may be learning something!
I found this page, which shows how to edit the registry to force "Allow the computer to turn off this device" to be unchecked, even if it doesn't appear on the card's Properties.
The post says that my Network Card should be Device Number 0001, and there should be a PnPCapabilities value of "24".
Turns out, my Network Card is Device Number 0002, and it does have a PnPCapabilities value of "24".
BUT.
But, my VPN shows as Device Number 0001, and does NOT have any "PnPCapabilities" listed.
And, turns out, the last two times my network's dropped, I've been tunneling through my VPN.
So.
SO.
So, I am creating the value, and keeping my CsEnabled at 0, so I have "Sleep" options in Power Settings, and we'll see what happens.
(The network card is a Marvell AVASTAR, and some Googling shows a recent update fudged everything up. As this is a new machine, I don't have previous drivers to roll to, but if this solves my problem, awesome. Besides, I'm more of a DC guy.)