"Hiring manager", refers to the manager, which is currently "hiring" for their department. Thus, we must assume that "hiring" is the present participle of hire. Then, the most fitting translation of this term would be:
einstellende Führungskraft
The disctinction between noun and present participle is important. The noun "hiring" would refer to the "the act of starting to employ someone" (https://dictionary.cambridge.org/dictionary/english/hiring). Then, "hiring manager" would have a different meaning, since the manager now manages the acts of starting to employ someone (cf. a human resources manager manages human resources and a logistics manager manages logistics). In the same sense the german "Einstellungsmanager" does not work.