Wouldn't it be an Intelligence (Religion) check, as you are a divine spellcaster.  

Quote from EnchantedNinja12345 >>

Wouldn't it be an Intelligence (Religion) check, as you are a divine spellcaster.  

No, in fact neither skill would be applicable. Arcana measures how good one is at remembering occult and planar lore, whilst Religion is the same thing but for gospel and religious doctrine. Neither measures one's prowess as a spellcaster, that would be one's spellcasting ability.

Also, interesting choice that this item, whilst it can be used as a spellcasting focus, cannot be used as a holy symbol. It does most of the things that your holy symbol would do, except it cannot trigger Channel Divinity, nor any other effects that require a holy symbol in particular.

The censer itself is “adorned with holy symbols,” so it can be used for anything for which a holy symbol is required. And from the PHB:

https://www.dndbeyond.com/sources/phb/using-ability-scores#Intelligence

Arcana. Your Intelligence (Arcana) check measures your ability to recall lore about spells, magic items, eldritch symbols, magical traditions, the planes of existence, and the inhabitants of those planes.

Quote from IamSposta >>

The censer itself is “adorned with holy symbols,” so it can be used for anything for which a holy symbol is required.

It doesn't say that in RAW, but if you intended it to work that way, that's an easy fix. You can simply say, "You can use the censer as a holy symbol"

Quote from IamSposta >>:

https://www.dndbeyond.com/sources/phb/using-ability-scores#Intelligence

Arcana. Your Intelligence (Arcana) check measures your ability to recall lore about spells, magic items, eldritch symbols, magical traditions, the planes of existence, and the inhabitants of those planes.

Yes. With Arcana, you can identify what spell is being or has been cast, the story behind its creation, what have you. However, when casting said spell, you use your spellcasting ability.

I point this out because neither clerics nor paladins use Intelligence as their spellcasting ability, so they're not likely to have that amazing Intelligence scores, and by extension, Intelligence(Arcana) modifiers. Using the wielder's spellcasting ability instead would make the check much more likely to succeed, and also much more appropriate to the flavor, seeing as how you explicitly don't know the cantrip, which you would need to in order for Arcana to be applicable. You can't recall something you don't already know.

Quote from CoolerKoolaid>>

Quote from IamSposta>>

The censer itself is “adorned with holy symbols,” so it can be used for anything for which a holy symbol is required.

It doesn't say that in RAW, but if you intended it to work that way, that's an easy fix. You can simply say, "You can use the censer as a holy symbol"

It quite literally says that RAW, it’s the very first sentence in the item’s description: “This antique, bronze censer is adorned with holy symbols.”

Quote from CoolerKoolaid >>

Quote from IamSposta >>:

https://www.dndbeyond.com/sources/phb/using-ability-scores#Intelligence

Arcana. Your Intelligence (Arcana) check measures your ability to recall lore about spells, magic items, eldritch symbols, magical traditions, the planes of existence, and the inhabitants of those planes.

Yes. With Arcana, you can identify what spell is being or has been cast, the story behind its creation, what have you. However, when casting said spell, you use your spellcasting ability.

I point this out because neither clerics nor paladins use Intelligence as their spellcasting ability, so they're not likely to have that amazing Intelligence scores, and by extension, Intelligence(Arcana) modifiers. Using the wielder's spellcasting ability instead would make the check much more likely to succeed, and also much more appropriate to the flavor, seeing as how you explicitly don't know the cantrip, which you would need to in order for Arcana to be applicable. You can't recall something you don't already know.

 
You don’t have to recall the spell itself. As you’ve pointed out, you don’t know the spell, you cannot recall something you don’t know. You do have to recall “lore about the spell” to figure out how to cast it. Recalling lore and figuring things out most assuredly falls under the purview of Intelligence (Arcana). And the DC is only 10, so with absolutely no bonuses whatsoever that’s still a better than 50/50 chance of success, 55/45 to be precise. Even with Int as a dump stat, that still leaves a flat 50/50 chance of hitting the DC, presuming either standard array or point buy. That’s very achievable.

Quote from IamSposta >>

 
You don’t have to recall the spell itself. As you’ve pointed out, you don’t know the spell, you cannot recall something you don’t know. You do have to recall “lore about the spell” to figure out how to cast it.

I've just had a revelation: What if this whole debate doesn't matter, because the function simply shouldn't require an ability check at all? Think about it, cantrips are the least powerful spells you can cast, akin in raw effectiveness to the Attack action. You already can't use the property more than once per long rest, I'd think that's balance enough, especially for casting such simple spells.

Compare this to the hat of wizardry. If a Wiz needs a DC 10 check, then everyone should need one. 🤷‍♂️

Quote from IamSposta >>

Compare this to the hat of wizardry. If a Wiz needs a DC 10 check, then everyone should need one. 🤷‍♂️

I don't got Xanthar's :(

It’s similar to this, just for Wizards.