Sharing my experiences with a Peteran (3 Ambitions and one free-play Kingdom in), it seems that Peterans increase most easily in popularity early in play, when there are fewer sources of entertainment or venues for town Sims to visit. Even so, Peteran popularity increases more gradually than Jacoban Fear. On the flip side, there is no penalty for being too popular. That said, here are my tips:
Try to hold two sermons a day, and schedule at least one evangelism session on the monastery platform between them. This will prevent popularity from decaying dramatically between sermons.
At Level 3, Peteran Priests unlock the "Inspired" sermon tone by Studying the Watcher at the archive. A full hour of study isn't necessarily required, but the "inspired" buff does take about 35 to 40 Sim minutes to be acquired through study. It will last until the next sermon, but the Priest will have to Study the Watcher again to renew it, in order to give another sermon with the "Inspired" tone. Inspired sermons seem to be more effective overall.
When giving a sermon, pay attention to the responses of your congregation. If Sims are booing or heckling your sermon, they are unhappy with the tone and it is likely to be less successful. Change tone until you find the one that makes them laugh to themselves. This represents a favorable response (even to Solemn toned sermons).
Convert as many locals as you can. A sermon will usually attract at least one non-Peteran Sim to the Monastery lot. Easy way to spot them is to click each congregant while your priest sermonizes, and select the "Peteran Priest" interactions from the menu wheel. If they are Jacoban or Agnostic, your priest will have a chance to Convert to Peteran Faith. I try to have a conversion interaction lined up to immediately follow my sermon, and sometimes you can get two or even three Sims converted before they leave the lot.
It is also worth converting Sims your priest encounters away from the Monastery. If your Focus is even half-full, a conversion is likely to succeed even at Level 1.
I do not know if making a lot of friends with your Peteran Priest necessarily improves the popularity of the Faith, but it certainly can't hurt.
Likewise, the "Pray With" interaction probably helps build loyalty to the faith as well, but this metric, if it exists, is hidden from the player. I only infer it to exist.
A good sermon gives a hero Sim who attends an 8 hour Focus boost, at 3x the Priest's level - i.e. a +12 boost from a sermon from a Level 4 Priest, maxing out at +30 for a good sermon at Level 10. I have not had a chance to see whether bonus levels beyond 10 also increase the value of the boost from a sermon.
The "Watcher's Favor" interaction seems to build relationship and may affect faith loyalty, but be careful with this as it can drastically cut into your Priest's rest. Annex Effenmont and stock up on the superior Energy Potions before you really cut loose with this.
Regarding the well water, there is one Blacksmith Quest ("Cutting Edge") where Blessed Well Water is a required ingredient. The Peteran Priest is a supporting Hero in that mission. Unlike the Jacoban blades (Sword of the Watcher, Jacob's Tooth) which require Consecrated Well Water and Watched Water respectively, the Angel's Talon (according to its flavor text a Peteran blade) does not use Blessed Well Water in its creation.
Blessed Well Water cannot be used instead of Consecrated (the Jacoban equivalent) Well Water, or Watched Water (from the Village Shoppe) for forging the other blades.
