TEDESCO-Venice originally DEUTSCH from Austria? #Austria-Czech

Debbie Lifshitz
 

I've hit a brick wall.
My TEDESCO family lived in Venice at least from the early 1790s, perhaps before, but not long.
They appear on the initial Census of Jews circa 1898 as Todesco d'Autriche. After that they appear in later censuses as TEDESCO. All first names and ages match. The family definitely lived in the Ashkenazi ghetto and spoke Yiddish (as well as Italian).
On the other hand, Selig Goldschmidt of Frankfurt, whose daughter married a Tedesco, wrote in his autobiography that the TEDESCO family originally came from Baden and that their original name was DEUTSCH.
Problems: Jews were not allowed to live in Baden near Vienna before 1806 (Encyclopedia Judaica and The Encyclopedia of Jewish Life), therefore this is not where they came from.
Germany, of course has a region called Baden but I am not sure how to proceed given that this is a region and not a town. 
Also, if they did come from Vienna (rather than Baden), would there be any documentation?
Any suggestions would be greatly appreciated!
Debbie Lifschitz
Jerusalem

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