Hmm. That particular item is out of stock and discontinued (also halfway around the world), but I did do a little more snooping, which led me to this (being offered by the same place I got my K-lite system), which advertises "provide low-current flow with no automatic shut off (this is critical for some low-current-draw devices - like the 40mA kLite Qube)"

Spendy for a power bank, but I may give it a shot.