A famous priest has strong connections with holy and unholy sources. Angel, demon, heretic, or saint

Because some vampire showed up out of no where and said, "Hey, this is your god now. Tell anyone where she came from and I'll kill you."

Because that is the priest's job, of course. When a heretic shows up, you call for the priest, not the cobbler or the baker.

>angels and saints
Those go without saying
>demons
Because depending on how the setting works, demons may not be able to be killed on another plane. That's how D&D does it at least. So, the priest on the material plane knows the demons that have been summoned up and that he's put back down, and which have now marked him out for death.
>heretics
this is an odd one. How do you mean "connection"? Does he just know the name or by reputation, like every man in Europe knew who Martin Luther was, or do you mean know on a personal level? Assuming the latter, there's no reason for him to know any heretics personally that are not behind bars awaiting execution.

>Because he's a master of inter-planar diplomacy.

>The guy functions as a point of dialogue between the two sides. Both sides tolerate him because he's charismatic, intelligent and wise enough to be useful and entertaining.

I like these two, it seems fitting for the description given.

>Sounds to me like this priest is hiding his true nature and he's a much bigger player than he looks.
Which could very easily make this one true as well. Also leaves open the possibility of the priest working for either side, or playing them against each other to advance the goals of a third force.

The war between good and evil is a scam, and he's in on it.

The priest went through his usual nightly prayers before leaving. At this point he could recite them all in his sleep, so often as he had gone through this routine, though twenty-five years in the cloth would do that to any man. As he finished with an amen, he turned away and began walking out. Before he got there, a thunderous knock echoed through the church.

"I’m just about to leave, sir. Please come back tomorrow."

A strange, deep voice uttered back. "Father, it's... it's urgent." Whatever this voice was, it certainly wasn't human. It was deeper than was remotely possible for a human. The priest knew it right away.

"Ah. Balax, yes, yes, come in."

The priest opened the door, and the creature walked in. Standing at well over 9 feet tall, it had to stoop to walk through the doors. His horns scratched the top of the doorjamb, and the priest winced. He'd have to repair that yet again.

The demon looked down at the priest and smiled. Always unsettling, seeing a demon smile. If the priest were a lesser man he'd run in terror, but Balax was an old friend, one of many that had taken an interest in this little, seemingly insignificant man. Angel, demon, or anything in between, the priest always seemed to find them turning up at his door in times of trouble. Lucky me, he thought, before he began leading the demon to the confession booth.

The booth was quite a bit larger than many churches out there, which was not surprising given the priest’s strange “clientele.” The contractors gave him a funny look when he asked that it be made fifteen feet high, but it had proven invaluable over the years. The demon took his seat, and the priest waited a moment to catch his breath before sitting down as well.

"So then, what brings you here tonight?"

"Well father," the demon said, "I have, in a way, sinned."

"Well, naturally. Such is the nature of your kind. One can barely hold you accountable for doing evil. After all, you are made of the stuff, are you not?"

"I... it's not like that father. I've not done anything evil. Quite the opposite, really."

The priest looked at the mesh separating them. He took out his lighter and lit up a cigarette. Tonight was about to get a lot more interesting. "Whatever do you mean?"

"Well, I was hiding in my place. You know the one, I've told you about it before."

"Yes, yes, of course. The one that lets you watch the people on the street."

"Yeah, that's the one. Well, I was watching one day, as I do every day looking for victims. It was then that I saw... her."

"Her? Whoever could you mean?"

"She was a little girl. Couldn't be more than five. She was walking with her parents, when suddenly, a man came out of the alleyway with one of those gun things you humans are crazy about. Started yelling something, I couldn't quite make it out, and her dad, he, well, he..."

"It's okay, friend. Just say what you must say."

"Well, the dad ran at him. I heard a couple of loud bangs, and the dad was laying on the ground, dead. The mother was screaming, and this man, he shot her too. I just... I kept watching."

"Well, certainly. It must have been interesting. You demons, you do prey upon the base instincts of man. There's no doubt in my mind that you enjoy-"

"No, father, that's just it. Seeing this girl sitting there crying, I felt something. Something bad, deep inside. I had to help her. So I ran from my spot, and I found the man as he was leading the girl to a, um, it's a van, that’s what you call those big boxes on wheels, yes? Yes, a van. Well I got to him, and I killed him, tore him to pieces. The little girl thanked me for saving her, and I just ran off."

The priest sat for a moment. He took a long drag from his cigarette, and parsed through what he had just heard. "So... you killed this man, then? To save the girl?"

"Yes, yes I did. And that's exactly the problem, father."

"I fail to see how that's a problem. You helped an innocent in her time of need. Granted you did it through murder, but still, an innocent life was saved today."

"Yes, father, but that's just it. What sort of demon am I? A demon does not save the innocent."

"Well, true as that may be, you did destroy a sinner. You punished the wicked, and is that not your lot in life? While you may have inadvertently saved an innocent, at the end of the day all you did was tear a wicked man to pieces. In my eyes, and I don't know how much this means to you, that is the most demon-like thing you could have done."

The demon gave an exasperated sigh, almost comical coming from such a fearsome creature. "Thank you, father. I needed to hear that, it really helped. I'll see you again next week."

The priest put out his cigarette in the ash tray, and gave a slight exhale and a nod. "Yes, yes, of course. Now go home and rest."

The demon was already gone, simply vanished from the room. Such a bad habit of his, but at least he had the courtesy to knock before coming inside. The priest got back up, and began moving again to the door to finally go back home. As he was walking, he saw a light begin glowing behind him.

"Father, I need your aid." The voice was both beautiful and terrifying as it echoed through the high ceilings of the church.

The priest stopped in his tracks. "Sariel, yes, of course." He lit another cigarette before turning and facing the angel. "Right this way."

Because he's pic related.
Or what user posted in , , and

Take the Good Omens route.
That being the big players are so busy with moustache twirling and ineffabillity to pay much attention to the beings on the ground.
After long enough they got to know each other and reached an understanding of sorts, and all the higher ups in cults and churches along with the beings actually doing the grunt work play by the same book, making the whole thing look good to their bosses while keeping the plate spinning.
Cult forms, priest hires pc, adventurers bust it, capture the leader who is broken out of the transport and repeats all over.
Because nobody wants to be caught inbetween the cold divine war turning hot in their backyard, and both sides believe that they are winning and so are content to play the long game.

Demons are not fallen angels, demons are the hybrid offspring of fallen and humans, when fallen are completely corrupted by sin they are called devils, people make the mistake of thinking 'devil' refers exclusively to Satan or Lucifer because he is often referred to as 'the devil' but its like the saying 'you're the man', because he is THE devil that everyone knows.

/rant

Is his name John Constantine?

Because he made evil his bitch.