A significant obstacle to doing an online search for Fritz Peter is that both Fritz and Peter are common as German first and last names. The German Wikipedia provides some basic pointers, such as his 1923 dissertation in Göttingen.
Arne Schirrmacher, "Establishing Quantum Physics in Göttingen: David Hilbert, Max Born, and Peter Debye in Context, 1900-1926." Springer, 2019, p. 87:
Table 5.1 Dissertations concluded under Born's supervision in Göttingen 1921-1925
Peter, Fritz Über Brechungsindizes und Absorptionskonstanten des Diamanten zwischen λ 644 und 266 1923 Z. Phys.
Note that the title of the dissertation differs slightly (266 vs 226) from the published paper based on it, as can be seen from this scan of Z. Phys. It is:
Fritz Peter, "Über Brechungsindizes und Absorptionskonstanten des Diamanten zwischen 644 und 226 mμ", Zeitschrift für Physik 15 (1923), No. 1, pp. 358-368
I assume mμ are milli microns as an old-fashioned way to refer to nanometers. The library of the University of Göttingen has Peter's dissertation available in paper and micro-fiche format. Since German dissertations typically list the birth place of the author ("aus ...") on the title page, it would probably be helpful to get access to this document. The online search mask for the library is at https://opac.sub.uni-goettingen.de . Select search for
[PER] person/author and enter
Peter, Fritz as the search term.
The German Wikipedia mentions that Peter was a student of Hermann Weyl, and Weyl's writings confirm this. For example:
Hermann Weyl, "Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics" [ed. Peter Pesic], Princeton University Press 2009, p. 165:
A number of years later, in Zurich, a pupil of mine, F. Peter, and I applied integral equations to the construction of a complete set of inequivalent irreducible representations of a compact Lie group.
Peter is also mentioned as one of Weyl's students on page 233 in the following paper:
H. Weyl, "Sur la representation des groupes continus", L'Enseignement Mathématique, Vol. 26, 1927, pp. 226-239
Le chemin que nous avons suivi, un de mes élèves, M. F. Peter et moi, met le problème de la représentation en relation avec la théorie des équations intégrales.
From the the scan of the paper
F. Peter and H. Weyl, "Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe", Mathematische Annalen, Vol. 97, 1927, pp. 737-755
we can see that Peter lived in Karlsruhe, a major city in Southwestern Germany, at the time of publication.
I cannot find anything about Fritz Peter's time as a teacher (of mathematics and / or physics?) at the Schule Schloss Salem, a private boarding school in Southwestern Germany near Lake Constance. The school was founded in 1920 by Kurt Hahn with support of Prince Maximilian of Baden. There is a Kurt Hahn Archive which keeps records of the school's founder and apparently also the school itself:
In addition, it holds a considerable amount of documentation about Schule Schloss Salem, including the estates of Salem teachers and information about other organizations which have followed Kurt Hahn’s educational ideas.
A conference record from a mathematical conference held in Tübingen in Southwestern Germany in 1946 lists Fritz Peter as follows:
Peter, Fritz, Stud.-Dir. Dr., Überlingen (Bodensee), St. Ulrichstr. 36
Given that Peter died only three years after this conference, it would be a reasonable assumption that he died in Überlingen and that you might be able to access his death record at the local registration office ("Standesamt"), unless privacy laws prohibit that (e.g. not a family member, not enough time passed).
The small town of Überlingen is located directly on the shores of Lake Constance ("Bodensee" in German), about 10 km from Schule Schloss Salem. In the 19th century Überlingen was part of the Grand Duchy of Baden. From 1945 to 1949 it was in the French-occupied portion of Baden. Since the founding of the state ("Bundesland") of Baden-Württemberg in 1952 it has been part of that state.
"Studiendirektor", here abbreviated as "Stud.-Dir.", is a title for senior teachers at a "Gymnasium" (a particular type of German high school) who also have some administrative responsibilities. The principal of such a school normally holds the title of "Oberstudiendirektor" (modern abbreviation: OstD), one level up from the rank of "Studiendirektor". I am not very knowledgeable how quickly a teacher can rise to the rank of Studiendirektor, but I think it is about 15 to 25 years. This would very speculatively fix the start of Peter's teaching career as no later than 1930.
I found a potential further publication that might be attributable to the same Fritz Peter the question is concerned about. I base my hypothesis on a match of the author's name, the author's place of living (Freiburg in Southwestern Germany), date of publication, and the fact that it appeared in a publication dedicated to teaching in math and natural sciences. The fact that the author's name is not given as Dr. Fritz Peter raises doubt about this hypothesis.
Fritz Peter, "Geographische Ortsbestimmung durch Höhenmessungen."
Unterrichtsblätter für Mathematik und Naturwissenschaften, (Otto Salle, Frankfurt a.M.) Vol. 44, 1938, p.23
Unfortunately, only a snippet is available at Google Books:
Von Fritz Peter in Freiburg. / Zum eisernen Bestand der Aufgaben zur sphärischen Trigonometrie gehört die geographische Ortsbestimmung durch Messung der Zeit, der Höhe und des Azimuts eines Sternes oder der Sonne, wenn Deklination ...