I think this job requires a 5 axis machine, not 6, and that is only if you're going to cut the whole case out of a block of steel.
Aren't watch cases first pressed out in a die, machined where needed, and then polished?
If that's the case, a die (possibly progressive, I'm not sure) would have to be made, and that isn't cheap in the West!
Do you have the proper CAD files in order to program the case for a CNC to cut the geometry, or will you have a case digitized so someone can create that geometry that is so much needed?
Not every watch case is machined from a solid block, only the very simple ones.
I'm really curious about the method that you plan to use... This has to be determined first before you start poking down the wrong road, or you may be poking forever, or worse, spend much more money than needed.
I'd say generate a manufacturing plan, investigate info on cases such as this would be manufactured conventionally, and then head down that path.