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All of the dialects of German are written down. There are numerous "Mundart-Dichter" or "-Schriftsteller" who deliberately write in these idioms, to raise awareness, to popularise their use, and to try to raise their status in comparison to idioms like "Hochdeutsch" or the "Amtssprache". See the BAP example above, or this:
http://www.comedix.de/medien/lit/mundart_sammelband_schwyzerduetsch.php
There are also plenty of dictionaries like this one:
https://www.schweizerdeutsch-lernen.ch/blog/schweizerdeutsch-woerterbuch/
It's certainly true that there aren't official spellings for most and perhaps all dialects. I don't know what the state of Swiss legislation is on that count, i.e. whether Schwyzerdütsch has any kind of protected status.
Oliver Schick-
All of the dialects of German are written down.
Glaswegian is written down, but it's written down phonetically, as there's no 'official' spelling. Same with Geordie. Never ask a Geordie how to spell 'dut' (hat) as a fight will start shortly afterwards. Definitely don't ask what it means, as a riot will swiftly ensue.
It's certainly true that there aren't official spellings for most and perhaps all dialects. I don't know what the state of Swiss legislation is on that count, i.e. whether Schwyzerdütsch has any kind of protected status.
It doesn't, at least not in the Académie Française sense. I started reading a Schwyzerdütsch dictionary (yeah, go me), and it had more possible spellings for each word than the number of words used in the definitions. It's madness out there, I tell you. Madness.
Brommers
I'll accept Hochdeutsch, but I'm assured that Schwyzrdütsch/Schweizerdeutsch isn't a written language, so you can just make the spelling phonetically as you go along. 'Es ist ein bitzli huggli', for example, if your chosen route proves to be a bit lumpy.