Well if it helps, "Too Fat" is basically just the Village Polka with new words. So play the Village Polka and let the audience draw their own conclusions. I usually do it as a medley with the Lindenau Polka, but you can string any number of similar old polkas together to make things interesting.
Some other German and German-adjacent folk/trad tunes that I've played on these sorts of gigs:
- Ein Prosit (this one is pretty much mandatory)
- Du Liegst Mir im Herzen
- Der Fröhliche Wanderer
- Clarinet Polka
- Freut euch des Lebens
- Trink, Trink, Brüderlein Trink
- Lustig ist das Zigeunerleben
- Zillertaller Hochzeitmarsch
- Waldeslust
And some of the more modern/poppy Oktoberfest tunes to consider:
- In München steht ein Hofbräuhaus
- In Heaven There Is No Beer
- Edelweiss
- Pennsylvania Polka
- Who Stole the Kishka
- Fliegerlied
- Country Roads (Very popular at the "real" Oktoberfest for some reason)
- Hey Baby! (Yes, the old Bruce Channel song. Also very popular in Munich, believe it or not, thanks to a modern remake.)
Now I could be reading too much into your post, but I get the sense that you might feel a bit of disdain for a lot of this music. Which I sort of get. Some of it is simplistic and not very musically interesting. Maybe even stupid. And definitely clichéd (I mean, that's the point, isn't it? That's what Oktoberfests hire accordion players for--the whole stereotype.)
But you know what? I like playing them! More to the point, I like playing them for people who like hearing them. Kids going absolutely ape$hit for the "Chicken Dance". People swaying back and forth with their beers to Edelweiss, singing along. Even people not really paying close attention to the music but nonetheless enjoying the whole atmosphere.
There are lifelong memories being made at these events. And there are troubles being forgotten, at least for an hour or so. How fortunate to get to be a part of that?