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.
4I would have liked to saw could be grammatical, but in that case it means "I would have liked to cut John into pieces", which is just slightly different. – Nate Eldredge – 2015-02-27T16:24:31.840
2
@Nate: True, but I think in the context of OP's question we can safely dismiss that as "a contrivance too far" :)
– FumbleFingers Reinstate Monica – 2015-02-27T16:29:02.960@200_success: I've rolled back your edit, because I specifically am not saying only one of the three versions is "acceptable". Two of them are okay by me. – FumbleFingers Reinstate Monica – 2015-02-27T17:50:39.260
1Would Garner have liked to have seen John sawn by Mary before he went to Canada, Mexico, and Greenland? – Jim Reynolds – 2015-02-27T17:57:28.253
@FumbleFingers Sorry for misinterpreting. I've answered separately. – 200_success – 2015-02-27T18:10:58.987
2Sorry, I don't understand. You say some people think it's more correct? But you quote Garner who says it's wrong? But underneath you say you think Garner's wrong. And there's no vetted grammar source anywhere. Sorry, FF, what are you saying? – Araucaria - Not here any more. – 2015-02-27T23:45:00.560
But isn't your grammarphobia excerpt actually supporting Garner's position? – F.E. – 2015-02-28T00:14:57.030
You also might want to comment on Garner's, er, use of the term *"a present-tense infinitive"*. – F.E. – 2015-02-28T00:23:22.397
@Araucaria: I did highlight my key point - in this sort of context I don't set much store by antediluvian grammarians and their pedantic prescriptivism. So far as I'm concerned it's a matter of *stylistic choice*. – FumbleFingers Reinstate Monica – 2015-02-28T13:03:19.097
@F.E.: I'm not about to start arguing the toss with the likes of Bryan Garner over grammatical terminology! My position is simply that if the BBC uses a particular construction in a learning English context, but Garner thinks there are "grammatical" arguments against that construction, it's pretty much de facto a matter of opinion which (if either) side is "right". My answer is purely intended to show that differences of opinion do exist (and to give some "justification" for the opposing perspectives), not to support one usage over another. – FumbleFingers Reinstate Monica – 2015-02-28T13:13:56.713