Follow-up. It was the battery. Ordered a replacement from amazon and it instantly returned to working state.

Of course in the intervening 4 days I already bought a got a used T3-V and I'm loving the higher resolution screen and generally upgraded feel of that, so I now need to find a new use for my T3 remote (maybe replace the Harmony remote in my gaming room with it and use the T3-V for theater :>)