Well, while the Autoloader's firmware was up to date, the actually LTO-6's wasn't. I gave that a shot and it now it's working!
So...perhaps maybe it was a Windows 2012R2 compatibility thing? I dunno.
Anyhoo...for anyone that ends up using an HP product with the LTO-6 Ultrium 6250 drive, make sure you've got its firmware updated (currently build 352W).