Refractory inclusions [calcium-aluminum-rich inclusions, (CAIs)] represent the oldest Solar System solids and provide information regarding the formation of the Sun and its protoplanetary disk. CAIs contain evidence of now extinct short-lived radioisotopes (e.g., (26)Al, (41)Ca, and (182)Hf) synthesized in one or multiple stars and added to the protosolar molecular cloud before or during its collapse. Understanding how and when short-lived radioisotopes were added to the Solar System is necessary to assess their validity as chronometers and constrain the birthplace of the Sun. Whereas most CAIs formed with the canonical abundance of (26)Al corresponding to (26)Al/(27)Al of ∼5 × 10(-5), rare CAIs with fractionation and unidentified nuclear isotope effects (FUN CAIs) record nucleosynthetic isotopic heterogeneity and (26)Al/(27)Al of <5 × 10(-6), possibly reflecting their formation before canonical CAIs. Thus, FUN CAIs may provide a unique window into the earliest Solar System, including the origin of short-lived radioisotopes. However, their chronology is unknown. Using the (182)Hf-(182)W chronometer, we show that a FUN CAI recording a condensation origin from a solar gas formed coevally with canonical CAIs, but with (26)Al/(27)Al of ∼3 × 10(-6). The decoupling between (182)Hf and (26)Al requires distinct stellar origins: steady-state galactic stellar nucleosynthesis for (182)Hf and late-stage contamination of the protosolar molecular cloud by a massive star(s) for (26)Al. Admixing of stellar-derived (26)Al to the protoplanetary disk occurred during the epoch of CAI formation and, therefore, the (26)Al-(26)Mg systematics of CAIs cannot be used to define their formation interval. In contrast, our results support (182)Hf homogeneity and chronological significance of the (182)Hf-(182)W clock.