To begin with the end of my searches, it seems that the most frequently recommended book covering Klezmer theory (among other Klezmer topics) is "The Compleat Klezmer", by Henry Sapoznik.
Here is the blurb from Amazon:
(Tara Books). This book is the definitive anthology by the world's foremost authority on klezmer music. Features an in-depth musical analysis, discography, 33 annotated melodies, historical background, annotations, resources, bibliography and extensive instruction on klezmer music. A MUST for Klezmer fans.
But some interesting resources also came up along the way.
As a starting point, Wikipedia has an article on Klezmer with a section on the Melodic Modes. That section points to another Wikipedia article on Jewish Prayer Modes, which are indicated as the source of the Klezmer modes. That section, in turn, relies primarily on Baruch Joseph Cohon's "The Structure of Synagogue Prayer Chant" (Journal of the American Musicological Society 3, no. 1, 17–32, 1950).
Cohon's primary citations are A. Z. Idelson's Thesaurus of Hebrew Oriental Melodies, volumes VII and VIII. Here is the description of the Thesaurus from the Jewish Music Research Centre:
The most important anthology of Jewish music. The first five volumes contain music of Oriental Jewish communities which Idelsohn heard in Jerusalem during the years 1907-1921. The other five volumes contain music of Ashkenazi communities which Idelsohn collected from different sources. Volumes VI and VII are based on the manuscripts of the Birnbaum Collection at the Library of the Hebrew Union College Cincinnati.