Rabbi Moshe Rappaport? Katz Ashkenazi, [Dayan Frankfurt] - Surname Rappaport

I found this about the Rappaport family:

“The earliest known Rapaport is Rabbi Yaakov HaKohen of Porto, who is known by the Rapaport name from 1462. It is likely that his immediate predecessors fled Ashkenaz, Germany around this time which was the year of the expulsion of Jews from Mainz. The region of Southern Germany left was known as Rapa. They came to Porto, a river city in the Padua region of Northern Italy. Thus the name Rapaport tells of the families' geographical origins.

Many great rabbinical leaders and scholars, intellectuals and statesmen are found in the families' lineage. Among them are Rabbi Meshulam Yekutiel HaKohen of Rafa, the publisher in 1472 of the first Hebrew book ever printed. Rabbi HaKohen Rapaport was the light of the exile, serving as Av Beit Din, head of the Rabbinical court in the descendants of "The Shach". Rabbi Shabtai HaKohen wrote Sifte Kohen (1622-1663) a major commentary on the Shulchan Aruch, also formed a branch of the family.” (from Cohen-Levi.org)

Despite those last few lines, I believe Mr. Cherson is correct.

The book הש״ך ומשפחת רפפורט by יעקב גליס published by נחניאל may help explain where/if the Rappaport family ties in to that of the Shach.