- Silent Prayer - Amida
I understand that saying Sim Shalom in the Amidah is somewhat tied to the Birkat Cohanim. What is the source of the minhag to say Sim Shalom at Mincha on Shabbat in Eretz Yisrael when no Birkat Cohanim is included in the the Amidah..
Your understanding of the connection between "Sim Shalom" and Birkat Cohanim is indeed correct. The Rokach, Rav Elazar Rokeach of Worms, (11th century Germany) says in the name of his Rav, Rav Yehuda Hachasid, , that when Birkat Cohanim is said , so is "Sim Shalom". So is the opinion of Rav Meir ben Yekutiel Hacohen(13th century) the Hagahot Maimoniot who says in the name of his Rav, Maharam of Rottenberg, on the seder tefila of the Rambam that when Birkat Cohanim is not said such as by mincha, "Shalom Rav" is to be said. However, he also adds that at Mincha on Shabbat "Sim Shalom" is said because at Mincha on Shabbat we are after reading the Torah and in Sim Shalom we have the words "כי באור פניך נתת לנו תורת חיים" which refers to giving of the Torah. ('פירושי הרוקח' (סימן מא) ,הגהות מיימוניות סדר תפילות נוסח ברכת התפילה,א,) May you have a Chag Same'ach