Answering all this in no particular order, but starting with Gottii...
I'm not sure why you think I can't give a decent argument even if you haven't asked me or why any of this is exceptionally difficult to buy into. Still, it's almost certain that's a segment of the population Constantin preaches to (as he's intimated, he's very concerned with disenfranchised people all over the cluster). If you want to imagine that his entire congregation is made up of non-entities, then that's fine. There's a perfectly easy way to reach the Caldari population that really doesn't seem that complicated and, considering there are quite a few Caldari players working through this stuff ICly, it's definitely not a unilateral problem. If that's what you need to assume to squeeze it into your RP so that Constantin's ministry isn't shredding your character's reality for the State, then by all means just go with that. It's not essentially important who specifically Constantin is reaching with the ministry, only that he's doing it and the way he's doing it. I'm just worried that ministering to the non-entities is going to give Constantin a lot more political power than I really wanted him to have. He's meant to integrate and operate in these societies, not begin carving a feifdom out of the cluster. He's very, very obviously not an executive authority, he's a minister. It might be easier to earn Caldari ire by essentially taking over their dredges and creating a loyal workforce out of them than by trying to work with the government so he didn't step on toes. You guys are the experts, though, so if you want me to play that angle, I can.
As to Christian background, I'm not a Catholic and haven't been to church since I was six. Considering they use the same ecclesiastical titles and New Latin sounding language, I'm not sure how "not Christian" we're supposed to make the Amarrian religion. CCP didn't spawn it in a vacuum as original content. Before I played an Amarrian priest of any kind, I looked over the Amarrian religion stuff as well as doing some background research on Catholic practice to back up my RP. CCP doesn't give you a ton of details, so it's up to us to fill in the gaps. I actually was trying to incorporate some East Orthodox into it, but I didn't think it would gel with the people already using Catholic titles and duties, like Gaven playing a cardinal. I had to make sure he and others outranked me, but it was very important that Constantin is a bishop. It's the same as us essentially figuring that a megacorporation works like a corporation here on earth. We've got every indication that they do, they use the same titles so far, so it stands to reason that we can assume a CEO and board operate the same way than figuring that the CEO needs to consult the janitor just because we don't have that specific information. If it's really bothering people that I'm filling in the holes with historical background rather than creating a completely idiosyncratic rank or job nobody would recognize, it might be best to know what information I am allowed to include and what I'm not. We're obviously getting most of this stuff from real life and history if it's not spelled out, and a lot is not spelled out.
To fill everyone in on the basics if they're curious, bishops have some authority, but they're definitely not anywhere near being a duke or viscount. You could make that case for an archbishop and definitely a cardinal, but I wouldn't usually figure that with a plain-jane bishop. Despite being a cool-sounding title, they're essentially the first step of administrative authority above a church level. All the higher senior functions, those you'd probably attribute to a marquis, are much higher order bishops.
If you're trying to match his rank up to a traditional feudal lordship, I suppose bishop translates best to baron. He's really not very high on the ecclesiastical food chain, despite how stunningly huge his territory is. For modern comparison, a bishop is nominally the authority over a diocese. The entirety of the Catholic world is divided into a little over 2200 dioceses, averaging about 550,000 parishioners per diocese. I figure that proportion pans out, but your church leader isn't a legal authority. I've made sure Constantin truthfully assures people he isn't an executive authority. There is a part of the Theology Council that handles that sort of thing, but I figured that a bishop can't overrule the Privy Council's purely legal representatives in the area. I get the impression that they have a fairly limited executive role, centered wholly on heresy over regular crime. Constantin isn't even that, he's a minister. His role is purely ecclesiastical.
I hope that gives you a sense of scale and purpose. I'm not an expert, but I can go over the ecclesiastical stuff I use as references. In the interest of giving people some background, as Silas has rightly pointed out that some people might not be following along with all the bits, I've included some spoilers....
[spoiler]
Trust me, he's not walking into these places and demanding anything, as you've probably noted from his manner. Even if he was Cardinal Constantin Baracca the other three empires don't owe him any kind of consideration, and he's outranked by cardinals by about seven or eight steps. As he's enumerated, time and again, he largely succeeds as a foreign minister by being likable, affable, and cooperative with local governments. He succeeds, but not by being confrontational or asserting his authority anywhere. Obviously he absolutely cannot afford to be seen as a foreign government operative, especially where the Caldari are concerned. Luckily, it's true on his part. Constantin's family has more influence and he has members of his family in the Theology Council in MUCH higher areas, but he himself due to his position and mandate has little direct political influence back home. He's on his own out there, a situation that isn't changing anytime soon.
Really, it's all a matter of scale. Constantin is succeeding beyond his superiors' wildest dreams (or nightmares), but it isn't like he's commanding whole planets or even whole territories. He might have a lot of followers based on force of personality, his tireless ministry, and his comparatively accommodating teaching style, but spread out across the other three empires entire populations, he's nowhere near a powerful enough political force to register in any one place. His ministry is spread out across the cluster in little patches and sometimes single individuals who might be the only ones in their entire region who need an officiate of the faith. Unlike a lot of other dioceses, most of Constantin's "diocese" is full of non-believers he is trying to convert, rather than being full of state-doctrinated parishioners. He's got his work cut out for him.
Simply put, Constantin is a minor officiate who has probably the biggest diocese in terms of raw population (because with that many people, it's easy to get that many followers listening in) but almost nothing in terms of population density. He actually really lacks rank and resources even not considering the vastness of space, and thus he has to sort of improvise, especially since the law isn't necessarily built to favor him in the space he ministers in.
The big question people should probably be asking is related to why the status quo stands. If the Theology Council outright hated him and what he was doing, they'd recall him or replace him. If they outright loved him and were ready to go full-force into his ministry, they'd promote him and give him more resources. If they loved what he was doing and hated him, they'd install someone over his head. But none of these things are happening. Constantin's had to improvise a diocese out of this situation and continues to sort of fend for his flock without any further help coming from the Council itself.
So those are sort of the central mysteries. If you want me to just tell you up front how he's pulling this off and how this relates to his superiors, send me an EVEmail.[/spoiler]