As stated in the comments, and unless a manuscript with the author's name is discovered, we currently have no way of knowing who authored that targum. In fact, already in Rav Hai Gaon's day the author's identity was unknown (!), as he wrote (Teshuvot Ha'Geonim, siman 248):
"אין אנו יודעים תרגום ארץ ישראל מי אמרו."
Translation: "We do not who authored the Targum of Eretz Yisrael."
However, in his time there were people who had a tradition that even some of the greatest of the Amoraim of Eretz Yisrael used it, making it an ancient targum on par with Targum Onkelus (!). Granted, though, it is unknown whether Rav Hai was referring to Pseudo-Yonatan or to what's now known as "Targum Neofiti".
According to this article in Mishpacha (which also brought the quote by Rav Hai), it is generally accepted that the author or authors were sages in Eretz Yisrael. As many sources over the generations called it "Targum Yerushalmi", it is believed these sages lived in Yerushalayim. The article brings a few opinions attempting to narrow down the authorship a little more:
Rabbi Bernard Revel thought it was created by Eretz Yisrael sages of the 8th or 9th century, as a reaction to wrongful translations inserted into targums by local Karaites (similar to what some believed they had done to the Talmud Yerushalmi - see here).
Rabbi Menachem Kasher disagreed with Rabbi Revel, being that there are many instances in the Targum that simply do not make sense as anti-Karaite measures, per Rabbi Revel's view, and secondly, that if such a targum project was started around Rasag's time, why is it not mentioned in any of his or his contemporaries' writings? And thirdly, there are sources (such as Rav Hai, brought above) that date it to much earlier.
Some scholars (nameless in the article) believe that the Targum was written no earlier than the late Savoraic-early Gaonaic period, being that there are instances in which the Targum mentions historical facts only relevant to later periods, such as the names of Yishmael's wives being the same names as Muhammad's wife and daughter (see Targum on Beresheet 21:21). However, the article points out that at best, this is evidence that there are later layers in the Targum, but it's not evidence that the entire work can be dated to a later period.
See here for a summary of the article in Hebrew.
Academic scholars nowadays seem to generally argue for later and later periods of composition, although the Mishpacha article's note on this (that this reflects later additions to the original material), in my humble opinion, is still valid:
Professor Stephen Kaufman in his essay "Targum Pseudo-Jonathan and Late Jewish Literary Aramaic", Aramaic Studies 11, argues that the Targum is an amalgamation of several different older targums, including Onkelos and Neofiti, together with chutz-la'aretz (outside of Eretz Yisrael) targums. For this reason, together with his deeper analysis in the essay, he surmises that the author lived somewhere between the 6th and 9th centuries, somewhere in the area of Eretz Yisrael or Syria (i.e., not necessarily in Eretz Yisrael), but it is difficult to narrow it down more. Similarly, in this essay by Dr. Iosef Zhakevich, several opinions are brought that state that the author of the Targum wasn't so much an author than he was a compiler.
Dr. Leeor Gottlieb in his essay "Towards a More Precise Understanding of Pseudo-Jonathan’s Origins", Aramaic Studies 19, argues for a much later dating of the work and a vastly different location. In his view, it was authored in 12th-century Italy (!!!). This is pretty much the view that Gavin McDowell suggests in his essay "The Date and Provenance of Targum Pseudo-Jonathan: The Evidence of Pirqe deRabbi Eliezer and the Chronicles of Moses", Aramaic Studies 19. He even goes as far as to suggest a person who would have had the necessarily tools and materials to compile such a work: Rabbi Menachem ben Shlomo, though he doesn't outright state that he believes that he was the author of the work. In his words:
"It could be the work of a single author. Menahem b. Solomon, for example, had all the necessary resources to produce a text like Targum Pseudo-Jonathan. First, he knew a Palestinian Targum similar to the one that must have served as the base of Pseudo-Jonathan. Second, he knew Targum Onqelos, which he cites in Sekel Ṭob. Third, he was conversant with the entire rabbinic corpus, not only the classical canon of Talmud and Midrash but also later works such as PRE [Pirkei D'Rabbi Eliezer]. He also seems to have known the Chronicles of Moses: While discussing the fourth plague, he refers to the silonit (סילונית), an extremely rare word which, in the Chronicles of Moses, designates a sea monster that ravaged Egypt during this plague. That an Italian author such as Menahem b. Solomon might compose an original Aramaic work at such a late date, although unusual, is not more so than the appearance of the Zohar in thirteenth-century Spain."1
1 It's interesting that the validity of such a late dating of the Targum hinges, according to McDowell, on the supposed late dating of the Zohar. It appears that strong-enough arguments against the late dating of the Zohar would also significantly weaken late datings of the Targum as well. See here and here for some arguments for the earlier dating of the Zohar.
