Importance of Distinguishing from House Indoril
Self explanatory. House Indoril currently has more of a aristocractic/executive air while the Tribunal Temple is the religion itself. Should also be of note that the Temple has a history of being charitable to the poor (Something Caius Cosades said was the major he reason he respected them), another possible distinguishing factor from the Indoril, who may be more interested in political advancement.

Distinguishing from Vanilla Temple
Corruption as a Focus
Despite how much this was touted in vanilla, you don’t get to see much of this portrayed in vanilla quests outside some mentions in the Main Quest. More focus on this could then play a part in TR’s characterization of the Temple, particularly as their power base exists on the mainland.

Almalexia, Sotha Sil Focus
Vivec lives on Vvardenfell, and so most of the Temple’s quests are based around his teachings in particular (Seven Graces pilgrimmage, etc). It’s proposed the mainland should then have more quests and focus on the other two members of the Tribune, Almalexia and Sotha Sil. The Vvardenfell Temple could then be understood as a Vivec-ian branch of the Temple.

Plotline Focus
Suggestion 1: The first part of the Temple questline is focused on familiarizing the PC with the religion. They make pilgrimmages dedicated to each of the gods, to the ancestors, and to each of the saints. They are then accepted into the priesthood, in which they are sent on pilgrimmages to do with more secretive teachings of the Temple, as well as a story-plot based around the doings of Necrom. (Whatever this entails?)

Suggestion 2: Temple quests run along three major themes, Mercy (heal the sick), Mastery (daring deeds), and Mystery (esoteric secrets).

Suggestion 3: Some of the pilgrimmages could be about rediscovering for Sotha Sil’s holy places.