Some people might think OP's original text using the Perfect Infinitive (to have seen) is more "logical" because it more explicitly echoes the past tense element of would have liked. But note this from Garner's Modern American Usage (2009)...
would have liked
...should invariably be followed by a present-tense infinitive — hence would have liked to go, would have liked to read, not ✳would have liked to have gone, ✳would have liked to have read.
Having said that, Garner himself acknowledges that the erroneous phrasings are very common. In fact, Google Books claims 386,000 written instances of would have liked to have seen. That's not so common as the "correct" version would have liked to see with 1,050,000 instances, but I think it's enough to justify saying that unless you need to pass a "fussy" exam, you could reasonably call it a stylistic choice.
OP's "Simple Past" version ✳I would have liked to saw is idiomatically and grammatically unacceptable.
EDIT: I see the question has been edited to tell us that the supposedly "incorrect" (according to Garner) version actually comes from a BBC Learning English page. I take this as further evidence (if it were needed) that Garner's position is unjustifiably pedantic/prescriptive. And comparing American and British charts on Google NGrams, I see no evidence that his position reflects any kind of AmE/BrE usage split.
At the risk of stoking controversy on what I consider to be something of a non-issue, I'll just cite this from grammarphobia, who also seem to have little time for Garner's position...
Using two “haves” (as in, “I would have liked to have gone”) is usually incorrect, because it’s unlikely that you really intend to talk about two separate times in the past.