I do not think that the father of Mordechai ben Shalom was Pinchas of Korets
Pinchas of Korets was not Shalom Pinchas. He only had the name Pinchas. He is never referred to as Shalom Pinchas of Korets. On his matzeiva he was called Pinchas ben Avraham not Shalom Pinchas. I would be interested if anyone has evidence to the contrary.
The tradition in our family is that Mordechai ben Shalom is descended from Nathan Nata Shapira the Megalleh Amukos (based on a written statement by Eliezer Elias Pinczower) but we do not have any tradition about being descended from Pinchas of Korets.