DebH, the North Face trail makes a long traverse from Schiltalp out to the southwest, then loops back. for a shorter, easier, but sill scenic walk, you can simply drop down into the valley from Allmendhubel (aim for the Sonnenberg mountain hut for the easiest path) and walk straight down to Mürren from there.
You can see the route on this little map—-just follow the yellow dots from the Allmendhubel play area down to Sonnenberg, and the green dots from there. This says 30 minutes from Sonnenberg down to Mürren.
https://www.sonnenbergmuerren.ch/
You are correct about the lack of “cover” in that area—-no place to hide to take care of nature’s call. Maybe save your appetite for Sonnenberg and plan on buying a snack there, in case they do not have the “pay” option for the restrooms.
The restrooms at Allmendhubel are in the funicular top station, and accessible without going through the restaurant, so one is free to use those without payment or permission.