After talking with LaCie (Seagate actually) tech support, here is what seems to be the problem.
When the computer is fully charged a signal is sent to the drive to enter sleep mode. I am unclear whether that is the OS or the LaCie driver doing that. My settings tell the drive to never go to sleep. So I believe it is the LaCie driver doing the deed.
The 1big Dock hub then keeps the power off (sleep mode) until there is access to the 16 TB drive. There can be access to other drives attached via daisy chain to the HUB, but those are not counted as access to the drive. As a result the power remains OFF to the computer. And the Hub blithely allows the computer to drift down to zero power as it consumes its battery.
This is a severe logic error on LaCie/Seagates part. They presumed that the computers need for power is determined by drive access. What they need to do is to monitor any activity on the computer, or battery power state and use that to decide when to turn power back on. However, they may not have access to that information.
Alternately, they could periodically turn power back on, and once again receive a sleep mode command when the computer is once again at 100% battery.
However they resolve it, this is a driver/firmware problem in/with the 1big Dock hub.
For users, until they fix this, a possible solution (as yet untested) is to write a script to touch the 16TB drive periodically on a schedule. That should trigger the Dock to turn power back on. Something as simple as an ls for a nonexistent file (with the result routed to DEV/NUL) should do the trick. Every half hour is likely sufficient.
I did request that they forward this to the programmers for resolution.