As I understand it, the "power" budget here, is to effect change. An eInk display is statically showing the arrangement of the beads.
So a mechanical watch would presumably be doing something to charge a capacitor, to get sufficient "power" in electrons, to push some beads around.
If you want to count seconds with more than 1 bead, I suspect no. If you want to count minutes with a single digit, as long as its a low bead count, maybe. if you were up at 5 minute granularity, I think its possible high quality changes could be pushed out.
I don't know about a watch but a power supply using kinetic energy would work such as a small lever that you could move. Kinetic power supplies sound nice but they are usually way more expensive that a battery even after you factor in the life time battery costs associated with powering the gadget. Plus it's inconvenient. It's way more practical to have a general purpose kinetic energy power source.
Here's an article that will give you some insight.
So a mechanical watch would presumably be doing something to charge a capacitor, to get sufficient "power" in electrons, to push some beads around.
If you want to count seconds with more than 1 bead, I suspect no. If you want to count minutes with a single digit, as long as its a low bead count, maybe. if you were up at 5 minute granularity, I think its possible high quality changes could be pushed out.
You might be re-winding the clockwork a lot.