can't have been vs. couldn't have been

You're right; can't have been is strange.

For one thing, can is a modal auxiliary verb; all modal auxiliary verbs are irregular as hell.
For another, can't is a negative, and negatives with modals have different syntax.

Of course, a great deal depends on what the predicate of the sentence is.
Can't have been is just a string of auxiliaries, not a predicate.
It could have a predicate noun or a predicate adjective or a predicate complement

• They can't have been so drunk that they didn't notice their car had been stolen. (pred adj)
• They can't have been doctors; they were wearing nurse's uniforms. (pred noun)
• It can't have been playing the wrong note that infuriated him so. (pred comp clause)

In each of these examples, note that the corresponding affirmative sentence is terrible:

• *They can have been so drunk that they didn't notice their car had been stolen. (pred adj)
• *They can have been doctors; they weren't wearing nurse's uniforms. (pred noun)
• *It can have been playing the wrong note that infuriated him so. (pred comp clause)

This is because the epistemic sense of can is a negative polarity item,
and requires a negative trigger. Without a negative, you can't use can epistemically.

This is a fact like the fact that epistemic will ('be scheduled to, be sposta', sometimes called "future tense") is not allowed in hypothetical clauses (*If it will rain tomorrow, the picnic is cancelled).
Very strange syntax and many irregularities with modals.

All right. I've done some research concerning can't have been vs. couldn't have been.

To remind meanings and uses of couldn't have :

• it is used in past hypothetical conditionals - can't have can not be substituted
• expresses impossibility in past time
• embodies the root meaning (ability or capability) of the modals can and could - that they didn't have the ability or skill to.

Can't have and couldn't have share a similar degree of probability. They are equal in meaning when they express the impossibility of something.

The differences:

1. • "Can't have been" suggests that it happened more recently. It expresses a judgment about a recent action or situation, a context in which the issues are still fresh and relevant to the present.
     (e.g. Can you check my homework? I'm quite sure I can't have found every bug.)
   • "Couldn't have been" suggests that it was further in the past.
     (e.g. As a young woman who had never given birth before, she couldn't have known what the experience of childbirth would be like.)

2. Can't have is used to express negative obligation or permission, while couldn't have carries no such meaning. This use of can't have means "You are not allowed to have (written...)."
   (e.g. You have to write it during the class, you can't have written it beforehand.)

Though can't have is used much less than couldn't have.