I am certainly no expert on Nihongo, but I do have some familiarity with the Iga Ishido school. My two favorite swordsmiths are the Fujiwara brothers, Shizumasa and Shizutada. Shizumasa was the older brother and is ranked wazamono. He used several different mei, and used both 鎮政 and 鎭正 in his mei. Shizutada used several mei as well, but he always spelled his name as 鎮忠. Of course, the kanji in question is the 鎮. I’ve seen many, many ads that call them Shigemasa and Shigetada, but they still use the familiar 鎮, not 重. From what I’ve been told, before the Fujiwara clan migrated from Bungo, their name would sometimes be pronounced Shige. But from what I’ve learned, once they were established in Iga, it was always pronounced “Shizu”.
I’m certainly no scholar, so when I talk about the clan, I always use Shizu. But when doing web searches, I search for both Shizu and Shige.