I was collecting data for potential use on a 3D model of Kirika's pocket watch, when I realized the watch makes no sense.

The watch is a so-called "hunter case" as opposed to "open-face" watch. Open face watches have the stem had 12 o'clock, but traditional hunter-face watches have the stem at 3 o'clock. Only modern hunter-case watches sometimes have the stem at 12 o'clock.
The seconds dial is always at 6 o'clock on a traditional watch, except when a hunter-case movement is installed in an open-face watch, in which case the seconds dial can be at 3 o'clock. Or, as in the case of the "official" merchandise noir pocket-watch, a traditional hunter case with the dial rotated so 12 o'clock is at the stem and the seconds hand is also at 3 o'clock. I've found nothing about watches with the seconds hand at 7:30.
Note that the stem has no crown. This suggests that the watch is key-wound and key-set (making it very unlikely to be a modern watch). However, the little thingie at 11 o'clock is a set pin. It's a little button that must be depressed with a fingernail in order to set the watch, its purpose to prevent accidentally changing the time. It's a feature that only makes sense on a stem-set watch, but Kirika's watch has no crown on the stem by which to set the time.
And we never see a winding key. It should be on a watch chain to prevent separation from the watch.
It could be battery-powered and radio-controlled, of course. But then why the set pin? It seems they didn't put as much thought into the pocket watch as perhaps they should have...