I think the Lorsa 8FA is our winner! Neat! I agree with the lack of anything "Swiss" anywhere on it, and looks like I was right about the French looking shock spring. This will be interesting. The movement pictured is the one that's not supposed to be working, and is out of the one watch that's not quite like the others. I have no reason to believe the others won't have the same movement. <$2 for a French watch is pretty solid. Let's see how things shake out.
Work is done for the day, time to start hunting for more "494 Club" fodder!
I had an experience recently where a watch was experiencing similar problems. After MUCH looking I finally found the lower hole where barrel arbor rests was oval and barrel had lots of play. I happen to have a old movement so I switched the parts to the old movement plate and problem was resolved. So, I guess my reply is to see if barrel arbor upper/lower holes are worn and causing excess play and wobble.
Sadly those Accutrons don't show up at the 404 price point, but they are pretty tempting.
One day I may give in and pull the trigger on one anyway regardless of the price.
Meanwhile I'll keep looking, you never know.
If so that means it is probably French, rather than Swiss, which kinda goes with the styling, and the lack of "Swiss Made" anywhere.
Lorsa/ France ebauche are usually fairly good quality movements. I suspect you should have little trouble getting them running,and with a little bit of a clean they should look pretty smart.