One simple option to raise the profile of religion is to append a character's religion and their position in it to their message signature..
Severn Da Hadez
Count of Eurotan, Marshal of the Northern Guard, Sigrid's Warrior of Darkanism
I like the idea of having optional commandments similar to realm laws with a chance of getting caught breaking them dependant on the number of followers/temples/priests in the region. You would be reported to the Elders of the religion for transgressions. Priests could have a 'conduct inquisition' option where they have a higher chance of spotting transgressions occurring that day in the region.
Commandments could include things like resting on certain days, not duelling, not performing certain types of looting, not evading or retreating from battle and so on.
Nobles would have a 'piety' statistic. Religious elders would have a limited amount of marks (say 1 per member per week) with which to reward or punish members. The effects of good or bad marks on piety score would be related to how strict the religion is, making it easier for members to gain/lose prestige the stricter the religion's commandments are. Moving above or below certain piety thresholds would have honour/prestige effects for the noble
There is no reason the religion could not enforce additional commandments/requirements outside game mechanics though there would be no automatic reporting for those.
This would give players a tangible stake in their choice of religion and their adherence to it as well as adding depth to the religion and adding more potential for interaction between elders and members. It would also add another click-able thing for priests to do.
This could be added incrementally to the game (maybe add the piety system first then add mechanics for commandments),
