In what ways should we view this current position as a continuation of his previous stint on the job, and in what ways is this distinct? Does he need to say Shehechiyanu again since this is new? Do his previous years combine towards having seniority over other mods?
I see the precedent for this from Berachos 28a, in the reinstatement of Rabban Gamliel as Nasi of Sanhedrin:
אלא לדרוש ר"ג תלתא שבתי וראב"ע חדא שבתא
Rabban Gamliel should expound for three weeks, and then R' Elazar Ben Azaria for one
So, it seems Isaac Moses should only be a mod for three out of every four weeks.
ת"ר בית גרמו היו בקיאין במעשה לחם הפנים ולא רצו ללמד שלחו חכמים והביאו אומנין מאלכסנדריא של מצרים והיו יודעין לאפות כמותן ולא היו יודעין לרדות כמותן שהללו מסיקין מבחוץ ואופין מבחוץ והללו מסיקין מבפנים ואופין מבפנים הללו פיתן מתעפשת והללו אין פיתן מתעפשת
The Sages taught in a baraita: The craftsmen of the House of Garmu were expert in the preparation of the shewbread, and they did not want to teach others the secret of its production. The Sages dismissed them and sent for and brought craftsmen from Alexandria in Egypt, a large city with many experts. And those craftsmen knew how to bake like the members of the House of Garmu did, but they did not know how to remove the bread from the oven like they did. The shewbread was baked in a complex shape, and it was difficult to place it in the oven and remove it without breaking it. The difference was that these Alexandrians light the fire outside the oven and bake it outside the oven; and these members of the House of Garmu light the fire inside the oven and bake it inside. In the case of these Alexandrians, their bread becomes moldy over the course of the week, and in the case of these members of the House of Garmu, their bread does not become moldy.
כששמעו חכמים בדבר אמרו כל מה שברא הקב"ה לכבודו בראו שנאמר (ישעיהו מג, ז) כל הנקרא בשמי ולכבודי בראתיו וחזרו בית גרמו למקומן שלחו להם חכמים ולא באו כפלו להם שכרן ובאו בכל יום היו נוטלין שנים עשר מנה והיום עשרים וארבעה ר' יהודה אומר בכל יום עשרים וארבעה והיום ארבעים ושמונה
When the Sages heard of the matter that the bread of the imported craftsmen was of lower quality than before, they said: Whatever the Holy One, Blessed be He, created, He created in His honor, as it is stated: “Everyone who is called by My name, I have created for My glory” (Isaiah 43:7). In deference to God, the Sages should diminish their honor for the greater glory of God and let the House of Garmu return to their original station. The Sages sent for them to reassume their previous position, and they did not come. They doubled their wages and they came. Each day until then they would take wages of twelve maneh, and today they take wages of twenty-four maneh. Rabbi Yehuda says: Each day they took twenty-four maneh, and today they take forty-eight.
So basically, you should double his wages.