Put a 0.2 - 0.25 second on-target cooldown on Black Cleaver's passive armor shred.
I'm assuming the fact that Lucian's ultimate wreaks havoc on his enemies' armor with Cleaver is a contributing factor to the item being so strong on him. Not the only reason, mind you, because good AD + Rage + 20% CDR are all good reasons too, but a reason nonetheless.
A bit of math that I did earlier showed me that The Culling at rank 1, which is 20 bullets fired over a duration of 3 seconds, deals physical damage roughly 6-7 times per second. That's an instance of damage every ~0.15 seconds, give or take. Basically meaning that The Culling by itself will fully stack Cleaver on a target within one second. (At rank 3, or 30 bullets within the same 3 seconds, the interval decreases to a mere tenth of a second, at which point Cleaver fully stacks within a little over half a second.)
Giving it a very short on-target cooldown for the armor shred effect prevents The Culling from driving up the stacks instantly. If you're worried that this might impact other users of the item:
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2.5 AS means you attack roughly every 0.4 seconds, so basic attacks wouldn't be affected by this change. (Unless you're Kog'Maw, but then, nobody builds this item on Kog anyway.)
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At level 16, Garen's Judgment deals damage every ~0.33 seconds, so that isn't affected by this change either.
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Physical damage DoTs, like Urgot's E and Talon's Q, as well as Wukong's ultimate, typically deal physical damage per second as far as I'm aware, so this change doesn't interfere with them either.
Basically no single source in the game deals physical damage even close to as rapidly as The Culling does at rank 1.
If this seems like an odd way to nerf an item, Tiamat and its upgrades have a similar under-the-hood nerf on them. Their passive Cleave has a very brief cooldown between activations, which exists solely because of Shyvana. Specifically, so that her dragon form Q doesn't proc Cleave on everything it hits due to how it functions. (I'm not sure if this functionality still exists or if they've since worked it out, but either way, there is/was precedent for this sort of functionality tweak to curb an outlying case.)
I'm not sure this would fully solve the "Korean Lucian" issue, but it seems like a decent way to help curb it without meaningfully tweaking either the champion or the item.