Saving configuration data for a watch face.

Ok, what you're after looks like it's similar to some other clock apps - for example https://banglejs.com/apps/?id=worldclock (there are some others, like https://banglejs.com/apps/?q=zone)

Those use a 'customiser' page on the app loader to let you configure them from the App Loader (https://www.espruino.com/Bangle.js+Custom)

You can check the code out at https://github.com/espruino/BangleApps/tree/master/apps/worldclock but basically:

• An HTML file that runs in the app loader lets you configure timezones: https://github.com/espruino/BangleApps/blob/master/apps/worldclock/custom.html
  
• They are written to a file on the watch into called worldclock.settings.json
  
• The clock loads the data using data = require("Storage").readJSON("worldclock.settings.json")
  
  

is Bangle.js aware of timezones and daylight savings?

It's not enabled by default, but if you install https://banglejs.com/apps/?id=widdst then yes it is