A good place to start when one is researching Swiss surnames is always the Register of Swiss Surnames, which has been available free online for a number of years at http://hls-dhs-dss.ch/famn/?lg=ehttp://hls-dhs-dss.ch/famn/?lg=e
It shows where in Switzerland there is a family with a given surname and the age range when the family obtained citizenship.This lets you narrow your search, but of course does not mean that your ancestor was in one of those particular towns :-)It also does not show locations where the family previously had been present but no longer has any members with citizenship.
That being said, for Bärtsch one finds
Surname Canton Community Citizenship obtained Place of origin Bärtsch GR Chur 1857 (Furna GR) Bärtsch GR Furna a Bärtsch GR Jenaz a Bärtsch GR Luzein a Bärtsch GR Saas a Bärtsch GR Seewis im Prättigau a Bärtsch GR Zizers a Legend: a = before 1800; b = in 19th century; c = in 20th century.
The surviving church books for virtually all of the towns in the Prättigau valley are available on microfilm through the LDS Family History Library and their branch Family History Centers.These church books generally run up to the year 1875 when civil registration took over recording all vital records.