This question 'Would have known' vs 'could have known' is one of the most interesting questions I have encountered. I did a research about its use.

I suspect that in the linked question "I don’t know how the killer could have known the key code, but I’m sure that’s when he must have got in", we could roughly paraphrase it as "I don’t know how the killer would have been able to know the key code, unless somebody else had told him", as suggested by Frank's comment. Here "could + perfect infinitive" expresses past counterfactual meaning.

The other important use of "could + perfect infinitive" expresses past factual possibility, as suggested by HostileFork's answer.

This question 'Would have known' vs 'could have known' is one of the most interesting questions I have encountered. I did a research about its use.

I suspect that in the linked question "I don’t know how the killer could have known the key code, but I’m sure that’s when he must have got in", we could roughly paraphrase it as "I don’t know how the killer would have been able to know the key code, unless somebody else had told him", as suggested by Frank's comment. Here "could + perfect infinitive" expresses past counterfactual meaning.

The other important use of "could + perfect infinitive" expresses past factual possibility, as suggested by HostileFork's answer.

According to Ilse Depraetere 2009, the function of have in sentences with could + perfect infinitive either communicates anteriority (as in the example in (1)) or contributes to the establishment of counterfactual meaning (as it does in (2)), which are the main functions standardly attributed to have in could + perfect infinitive constructions.
And the paper highlights a special use of could + perfect infinitive in which have performs neither of the two functions (as it does in (3)).

1. Michael scrutinised the window display in the hope of discovering what had so held the child’s attention. Surely it couldn’t have been the dresses.

It could be paraphrased as “surely it wasn’t possible for it to have been the dresses”.

2. “How long did you spend recording the album?” – “Probably about a month of recording and a few weeks of tweaking. If I’d had a deadline it could have been quicker.

It could be paraphrased as "If I had been given a deadline, it would have been possible for me to have finished the album more quickly".

3. How is it that the Secretary of Defence could have remained at his desk until the crash? Whose responsibility is it to relay emergency situations to him?

It could be paraphrased as "How was it possible for the Secretary of Defence to remain at his desk?"

I suspect that in the linked question "I don’t know how the killer could have known the key code, but I’m sure that’s when he must have got in", have belongs to the second category of its functions.
We could roughly paraphrase it as "I don’t know how the killer would have been able to know the key code, unless somebody else had told him".
Do I understand it correctly? And I get some examples below from the Internet, with my own explanations added.

Which of them could be substituted with "would have known" or "knew" without considerable change in meaning?
I suspect when "could have known" expresses a counterfactual meaning, they are more or less interchangeable with "would have known" in the following examples. When "could have known" evaluates the past possibility they are more or less interchangeable with "knew".
Please help me walk through them.

#1. "My name is Elijah," the man says, inspecting her with dark eyes. "Your son – Malachi, is it? – asked me to help you."
...
He continues, "Like you, I'm a vampire, although I don't know how your son could have known (I'm a vampire) when he asked me to help you."

Here the speaker is expressing surprise. We could roughly paraphrase it as "I don't know how it is possible for your son to have known..". I think have belongs to the first category. Using 'would' wouldn't fit with the context. And why not just simply say "I don't know how your son knew"? It seems neater.

I'm not quite sure about the function of have here. Is it possible to use 'would' instead? I would reword it as "I don't know how much I knew about the festival at the time".

I think have here expresses counterfactuality; the parents didn't know his bad results. Using 'would' would work, I suppose.

#4. That spring I met a girl, or more precisely she met me. I was what is usually described as shy. In truth I was already somewhat socially and emotionally retarded. I had never really even talked to a girl. While my non-user and “normie” peer group were out going to social functions and starting to develop into young adults, I was always loaded and making no attempt at emotional or social growth. I avoided facing my fears and insecurities by not facing the real world. I don't know how she could have known the only way I would have been able to talk to her, but she found it by offering to smoke a joint. This was heaven for me. I didn't have to do anything—the pot did it for me.

It seems to be a hypothetical scenario here. I suspect 'would' is a possible alternative for 'could' while 'knew' couldn't be used.

I think have expresses anteriority, as in #1. We could roughly paraphrase it as "I don't know how it is possible for Dave to have known..". Seems 'would' couldn't work here. Again, I would just say "I don't know how Dave knew that". Any different connotation conveyed?

I suspect that in the linked question "I don’t know how the killer could have known the key code, but I’m sure that’s when he must have got in", we could roughly paraphrase it as "I don’t know how the killer would have been able to know the key code, unless somebody else had told him", as suggested by Frank's comment. Here "could + perfect infinitive" expresses past counterfactual meaning.

The other important use of "could + perfect infinitive" expresses past factual possibility, as suggested by HostileFork's answer.

And I quote here some extra examples from the Internet, adding some explanations of my own, so I might have a better understanding.

Which of them could be substituted with "would have known" or "could know" without considerable change in meaning?
I suspect when "could have known" expresses counterfactual meaning, they are more or less interchangeable with "would have known" in the following examples. When "could have known" is used for evaluating the past factual possibility they are more or less interchangeable with "could know".
Please help walk me through them.

#1. "My name is Elijah," the man says, inspecting her with dark eyes. "Your son - Malachi, is it? - asked me to help you."
...
He continues, "Like you, I'm a vampire, although I don't know how your son could have known (I'm a vampire) when he asked me to help you."

Here the speaker is expressing surprise. We could roughly paraphrase it as "I don't know how it is possible for your son to have known..". Using "would have known" wouldn't fit with the context. And why not just simply say "I don't know how your son could know"? It seems neater.

I would reword it as "I don't know how much I could know about the festival at the time". I'm not sure if it's the intended meaning.

I think "could have known" here expresses counterfactuality; the parents didn't know his bad results. Using "would have known" would keep the intended meaning intact, I suppose.

#4. That spring I met a girl, or more precisely she met me. I was what is usually described as shy. In truth I was already somewhat socially and emotionally retarded. I had never really even talked to a girl. While my non-user and “normie” peer group were out going to social functions and starting to develop into young adults, I was always loaded and making no attempt at emotional or social growth. I avoided facing my fears and insecurities by not facing the real world. I don't know how she could have known the only way I would have been able to talk to her, but she found it by offering to smoke a joint. This was heaven for me. I didn't have to do anything - the pot did it for me.

It seems to be a hypothetical scenario here. I suspect "would have known" is a possible alternative for "could have known" here while "could know" would not work here.

I think "could have known" here expresses factual possibility, as in #1. We could roughly paraphrase it as "I don't know how it is possible for Dave to have known..". Seems "would have known" couldn't work here. Again, I would just say "I don't know how Dave could know that". Any different connotation conveyed?

Which of them could be substituted with "would have known" or "knew" without considerable change in meaning? Please help me walk through them.

Which of them could be substituted with "would have known" or "knew" without considerable change in meaning?
I suspect when "could have known" expresses a counterfactual meaning, they are more or less interchangeable with "would have known" in the following examples. When "could have known" evaluates the past possibility they are more or less interchangeable with "knew".
Please help me walk through them.