And he dwelt in the wilderness of Paran, and took for a wife Adisha, but put her away. And his mother took for him Fatima to wife, from the land of Mitzraim.
Based on the references to the wife and daughter of the Prophet Mohammed, it seems that Targum Yonatan must have been written after the mid-600s.
The parallel passage in PRE seems like the foundation for Targum Yonatan's text, which offers a brief interpolation into the relevant pasuk of the longer aggadic version. The same reasoning applies to PRE, which itself must have been written sometime after the mid-600s, plus some unknown amount of time until the same tradition is found in Targum Yonatan.
Again, the parallel passage in PRE provides a fuller version of the aggadah, including the unique element of something bound around the waist signifying social status.
But Targum Yonatan has a unique perspective on the origins of the Akeidah, describing a dispute between Yishmael and Yitzhak that diverges completely from the version in PRE.
Targum Yonatan shares the tradition with PRE that Hagar was set free, and was no longer handmaid at the time of her full and binding marriage to Avraham.
Yet Targum Yonatan also has a layer that only recognizes Hagar as a handmaid, not a free woman, an internal contradiction.
Conclusion:
As with so many of our classical texts, Targum Yonatan does not have a single author, and was not written or created at a single moment in time. The text accumulated over multiple generation, and likely through the multiple different processes in use across those generations. Here's one plausible scenario.
The standard picture of the Meturgeman is a non-canonical oral tradition during the Tannaitic period. From this inception a consistent, but still oral, text develops in the name of Targum Yonatan. This foundational layer would use the Palestinian (eastern) Aramaic dialect.
Further elaborations on the now-canonical text, and integration of additional midrashic traditions, likely occurred during the Amoraic period. These introductions would use the Babylonian (western) Aramaic dialect, accounting for the inconsistency (see The Aramaic Bible, Targum Pseudo-Jonathan, Genesis, Introduction pp. 8-12).
It seems reasonable to assume that the transition from oral to written transition occurred at roughly the same time for Targum Yonatan as for the Gemara, roughly the early 800s (coinciding with the adoption of low-cost paper manufacturing in Baghdad -- see Paper Before Print).
Among the last layers to be added seems to be the interpolation of material from Pirkei de Rebbe Eliezer, as in the examples above. Coming in the age of manuscripts written on paper, it may well occurred in the same manner as so many other works of the Gaonim and Rishonim: the sofer creates a fresh copy of a manuscript, including wide margins for the new owner's comments and additions. For a respected scholar, the next sofer can incorporate some or most of the marginalia into the next generation of the work.
Scholars provide a variety of datings for that work, but again the 800s seems a reasonable hypothesis. All we can say with certainty is that the closing of Targum Yonatan post-dates PRE, but with the bulk of the Targum having far earlier origins.