Sometimes players build aspects and traits into their characters that they don't actually
want to be challenged at every turn. Just like showing an interest in befriending an NPC doesn't always mean the GM has to kidnap or kill those NPCs, this character's faith may be something the player wants as a cornerstone of their character, something immutable. They might be perfectly happy with you turning the screws on another part of their character.
However... what you're saying sounds like this ain't the problem. The player called you a bitch and flipped you off when you presented the problem. So far his solutions have been those of a lazy cleric. Splash holy water, collect Sword of the Cross? Petulant and indignant when that fails. Complains that another PC would raise holy hell if he killed the child? Make it obvious that you don't see that as the only way out, and it's not your intention for the True Believer to fall so easily into "kill the kid, collect Sword of the Cross".
After his one good idea, he then goes to the range and gets all passive-aggressive and goes to texting on his phone. Now the player is disrespecting your time and the group's time. He is the bitch now.
I think the player might be getting upset their Holy Sword isn't coming super-easy. Honestly, sounds like the player might not have the temperament to properly play a holy warrior if he's constantly thinking self-centered (and petty) thoughts about his character.
Drachasor has the right of it IMO. The player wants to play a lazy fuckin' cleric, to toss holy water at the problem and go home for pancakes. Lazy clerics don't get Swords of the Cross. The player doesn't want to engage himself in the game like the other players. That's a shame, but hey, work with that. He doesn't have to be in the spotlight. A lazy cleric can be an excellent supporting role. It's just too bad that your plot right now kind of centers on his faith and you've discovered that he's a terrible holy man.
I certainly don't know your bitch player's religious tendencies if any, but my more suspicious mind wonders if he is playing a shitty priest to act out passive aggressive feelings towards religion? Playing a lazy cleric might be how he sees the faithful. He might be trying to make some misguided point. That too is okay, but you might want to sit down and rework his Aspects.
I would plan for a failure scenario. The kid gets possessed and twists into some kind of terrible soulless beast that the party can kill without remorse in case ApathyMan over here can't be arsed to care about his character's motivations or the situation before him.
EDIT: I disagree with Katarn. Be open and explicit about the situation with the player. Fuck immersion and player/character knowledge splits. Tell him exactly what the stakes are here.