I won't be original, but building violence between the two groups on small scale is probably the easiest to go. You probably should follow the following steps:
Stage 1: Insecurity
You need to stage some (apparently random) murders. They have to appear as random as possible, such that people would start to fear for themselves, and be suspicious of each other. You probably should include one or two feat with high impact: horrible killing of a (few) child(ren), for example, to make sure to get everyone's attention.
For that you need to get mercenaries, thiefs and killers to do it for you. You might want to impersonate some member of the Joothums while contacting those. In any case you should make sure they cannot link those to the Gurfans.
If you can, having economic problem at this stage would help. Issue with supplies, rumours of bad harvest, etc. People are usually more prone to fighting when the economy isn't good.
Stage 2: Religious Motives
While continuing your terror campaign, you should start to attack Schijka's members and offices. But still get on with the murders. The change should not be too obvious. Slowly, the ratio civilian/religious will move towards the religous. The main idea is to make people believe that a criminal group has been formed and that they are targeting the Schijka.
Depending on the importance of the Gurfans, you might want to attack your own church as well. If the 3 groups are equally important, you need to do it to let it clear where it comes from. If the Gurfans are a minority, you can spare it.
In any case, there should start to have some diplomatic discussions, with some Schijka (together with the Gurfans?) accusing the Joothums of harboring criminals. Due to their pacific nature, the Joothums will run some inquiries to see whether some of their extremists haven't gone crazy.
Stage 3: Rumors and Mass Control
You don't want the inquiries to proceed normally. So by spreading rumours, and placing a few agitators at strategic places, you should get the mob to attack isolated Joothums members. The mob does not have to comprise elements of any religion in particular at first, but as the attack on the churches intensifies, some members of the Schijka will join the revolt.
You have now built an insurrection. The main goal of that step is to avoid inquiries done effectively. Furthermore, by implying mobs, you reduces the chances of the elite to control the members of their faith.
Stage 4: Arming the Opponents
You should then proceed to provoke more and more mob furious attacks, on both side. At this stage, it is likely that the lowest members of the churches will get involved.
You might need to control yours. Kill all the members of your clergy who might take part in any mob gathering. Whereas you target the most pacific ones on the other churches.
To escalade the violence, you need to be able to supply weapons to both sides. That will increase the violence to a state of warfare. Stage mob attacks against the milicia/armies of the religions.
You want to set clear objectives. So you can get rumours/speakers that emphasize the differences between the churches, and even set a deshumanization of the members of the opposing church. This will increase the hate and help built something like a nationalism up. A few chocking events on either side would help. "They decapitated a 5-year old girl! They hanged a pregnant woman!" etc.
You might consider a staged discovery of the original criminal group at this point. You might get the actual perpretator killed, or not. But you should arrange that members of the Joothums are identified amongst them.
Stage 5: The Spark
At that point, the stage is ready for a fight. The religious groups hate each others, and violent demonstrations are organised by both sides. The time is probably ripe for a good-ol' spark.
There are several ways to do that. And you can see what suits the best, from the events and reactions before, but one idea could be to find a few motivated Schijkas, and help them intent to kill an important person from the Joothums church (yes, Franz-Ferdinand). You should help them such that they can really get close to their target, but not really escape, and that they can't link back to you. Whether they kill their target or not, is probably irrelevant. The main idea is to imply the highest members of the churches in what was, up to that point, a base-class insurection.
During the War
If everything goes "well", there should be an open war at that point. But you should not just sit and sip, but you should build your side, with at least the following points
- make sure that no member of your church gets involved (from his side), so eliminate the more vindicative and protect the pacific,
- continue to supply weapons and some assistance to both side: if they see you as their ally, or neutral but depends on you, they won't try to get you in the civil war,
- organise the help to all victims/citizens. You can set up refuges or hospitals run by your Church. At the end of the war, you will be known as the pacific and reliable side.
What do you need to make all this?
There are a few things that should try to do prior to starting, but in particular you should have
- a certain control of the economy,
- an extensive spy network,
- fanatic members of your side, who can be spared later (yes, SA),
- financial ressources (weapons, to pay your spy, or corrupt a few people here and there),
- little moral and ethic.
