For one thing, this would imply that clinical psychopaths aren't conscious, which would be a very weird takeaway.

But also, how do you know that LMs aren't empathic? By your own admission they do "mimic empathetic behaviors", but you reject this as the real thing because you claim that with enough testing you would encounter a failure. This raises all kinds of "no true Scotsman" flags, not to mention that empathy failure is not exactly uncommon among humans. So how exactly do you actually test your hypothesis?