Your mission/challenge if you take it is to make that the german dialect/language of German Swisterland is officialised and nationally- legitimised, even made officially a language of it's own. Like, schooling and official texts in it, and an official position on it being an independent germanic language of it's own.
The problem is that there is not "the" swiss german dialect. Rather there are dozends of different (but mutually intelligible) german dialects spoken in Switzerland. The most prominent are:
-Basel: It separated from the rest of Switzerland by the Jura mountains an oriented more closely to high German, compared to the rest
-Berne: It has much more French loanwords, as it borders the french speaking parts of Switzerland, its speakers tend not to pronounce "L"s and is rather slow which leads to its speakers being mocked as dim.
-Zürich: It is rather harsh and fast spoken. Due to the large immigration (both from other parts of Switzerland and from outside) it has also changed the most. It is generally disliked by other Swiss because of the economic dominance Zürich has.
-Eastern Switzerland and St.Gallen: It's main characteristic are very high and bright vowels (especially the a).
Additionally there are many more dialects being spoken in the mountains. Often they are highly divergent, e.g. the dialect of Vallis is difficult to understand even for other Swiss.
Now the speakers of these dialects rather cling to their own, identify with it and would loath the idea of having to speak and write an other dialect. Additionally trying to imitate the dialect of someone else (if not done in a joking manner) is often viewed as trying to usurp their identity, and thus not well liked by the speakers of said dialect.
Thus even the members of the privileged dialect won't like the idea of others speaking it.
The example of Romansh is illuminating: It consists out of four dialects, each being spoken by not more than twenty thousand people. To make the distribution of schooling article easier a unified language oriented on the two most important dialects was created. It was also soundly rejected by the people who didn't want to speak this "corrupted; language. When the canton told them that it would not continue printing stuff in the four dialects it was either "Romansh Grischun" or German, they declared that they would rather have their schools in German than in this abomination of a language.
Now, I know the saying about how a language is often a dialect with an army,a navy and state behind it, and wish open playing with it so. Maybe an official grammar and orthograph etc homogenisation and officialisation, a 'legal swiss german' (what would be the name?), perhaps some changes, neologisms brought, etc...
Indeed force would be a way to establish such a language. The problem is that none of the major dialects is strong enough to enforce itself against the others. Historically Berne has been the strongest city (the dominance of Zürich is a recent development), with Zürich the second, Basel, Luzern and St.Gallen tied for the third place. However even Berne would be much weaker than the combination of all the other parts.
Another problem is of course that someone would have to have the desire to enforce such a Swiss German language.
During the middle ages the written language was Latin, and no one really cared about what the people talked.
Up to the french revolution French was the language of the learned people, who would again not care about what the peasants spoke.
Afterwards literacy spread amongst the general population, which lead to Standard German becoming increasingly popular in the German Part of Switzerland - Similar to how standard French started to supplant the dialects in the French part.
During this period the Swiss German was looked down on by the upper classes as a rural and inferior form, which no one wanted to enforce.
Swiss German only became popular amongst the elites during the first world war, when they wanted to make it clear to their French and Italian compatriots that they were not German.
However by the twenties this was again largely forgotten and Standard German started to proliferate at such a speed that if was feared the dialects would be lost in a few decades (as it happened with the french dialects). It was only after the Nazis took power in Germany that, Standard German was purged from everyday life and remained only in the written language for the lack of an alternative.
I think your best bet would be to somehow prevent World War I (which stopped the acceptance of High German) for a while. If Germany then becomes "nasty"; (e.g. the Social Democrats win a majority, the emperor and the nobility annul the election and use martial law to break the subsequent strikes, shoot protestors and so on) or the war breaks out at a later time, the individual dialects would be weakened to much to be revived. Thus the lowest common denominator of the dialects (which still differs considerably from Standard German) could be implemented as a Swiss German language.
How would react the french and italian speakers, cantons, etc?
If the new language is spread without force they will rejoice because they can now learn one language and understand everyone and don't need to decipher the different dialects (a cumbersome task for a non native speaker).
If it would be done by force they would resist it because it sets a worrying precedent for them. Remember 65% of Switzerland speaks German. If you can force them to accept a new language it will only be small step to force this language on the remaining 35%.
(My apologies also, this was based on that I heard it is mostly a spoken language, that 'official german' is offered in schools instead (maybe), and that its used also by artists, writters, tv shows, etc... I wish to offend no ones.)
It is true that Swiss German is mainly a spoken language and now standardised orthography exists. This generation often writes their E-Mails/SMS/WhatsApp in Swiss German, making up the orthography as the go. However, you seem to underestimate how widespread its use is in the spoken language.
All the written material in schools is of course in Standard German, but the lessons themselves are mostly in Swiss German. Whether this is something good or not and if it should be changed is a subject of continued (and fierce) debate.
Swiss German is also used in Radio and TV most of the time, only news are read in Standard German (and of course any interview/talk show which non Swiss participant as well).
Most of the political discourse is also in Swiss German unless someone in from the French/Italian part is present.
