Please play a bit with the Austrian on-line telephone directory. Use "Erweiterte Suche" (= Extended Search), and put nothing or Steiermark into the field for Bundesland (= province).
You will find there are a few Beinstingl left today with telephone listings, of which one family lives in Styria, i.e.,Graz. Much more frequent spellings, quite concentrated in Styria, are Birnstingl, Bohnstingl, Bonstingl, and Ponstingl.
The names Klement (though frequent and rather widely spread) and Ehmann also point to Styria. Rheinberger (in this spelling) appears to mostly exist today in Vorarlberg, the westernmost Austrian province (which borders the Rhine river). Today's spelling of the name in the east is Reinberger.
It is hard to say, but indications are your ancestors might have been from Western Styria (area of Lannach to Stainz), or alternatively from Eastern Styria (Fürstenfeld area) which would be close to Burgau which you are searching as well. Graz itself is also an option.
What is the reason why your ancestor shows up in a book coming from the order of the Benedictines?