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tl;dr What you describe is absolutely allowed (using a bivouac to spend one night). If you set up a small camp (described as a "planned bivouac" further in this text), you shouldn't be in a protected area. Bivouac Wikipedia Sleeping one night without a tent or a small igloo. An emergency bivouac is basically allowed everywhere. A planned bivouac ...
The linked topic of @Eyal is a good starting point to get an idea what to expect on those kind of hut treks. I would recommend to get information from online trip reports like on the site you linked to. They already marked the different sections of the Via Alpina where you could pick out 5 sections/days. First of all you should get something to work with ...
With the information on the website and g00gle maps, I think the whole parcour should be between 1500m and max. 1600m.
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