What do traditional sources say regarding the date of authorship of Midrash Tehillim?
Buber says there are two traditional attributions. One is by Rabbi Avraham [sic] HaSefardi in Ohel Yosef, where he attributes it to Rabbi Yochanan (the redactor of the Talmud Yerushalmi). However Buber himself must be making an error, as the author of Ohel Yosef (published uncensored under the name Tzefas Paneach - a famous and controversial pirush on the Ibn Ezra) was written by Rabbi Yosef Ben Eliezer HaSefardi. Given that this is sourced to a specific verse in Vayikra, I think that is who he means.
The other one, which is traditional in the sense that it is made by a Rabbi within the Jewish tradition rather than an academic, is by Rabbi Aharon Moshe Padua of Karlin (author of the ביאור הרא"ם to Medrash Tehilim) where he says it was (started by) Rabbi Shimon the son of Rebbi Yehuda HaNasi, by essentially making a Diyuk in the Talmud that he says hints to it.