The perfect infinitive (have + past participle) is necessary to mark the event as past, but there are two possibilities - both the regret and the event are past, or the event is past but your regret is still present. When you want to make this absolutely clear, then the two forms aren't alternatives but are a necessary choice :
I didn't want to do it at the time, but now I would like to have done it. (present regret about past event)
I would have liked to do it at the time, but now I'm glad I didn't. (Past regret, past action)

However, in the second case you could, from a grammatical point of view ( and should from a logical point of view) also say I would have liked to have done it. as both the regret and the event are past. But it's a form which is rarely heard. Why?

English as a rule hates marking meaning grammatically twice - think for example of the way it avoids double negatives (I don't want any rather than I don't want none) and the almost complete lack of marking of person in verbs. The subject indicates the person so verb inflection is not necessary - it would be a second indication of the same thing. The third person singular is an anomaly in this respect.

In your example, we have to mark the verb phrase as expressing a past event, so we need at least one of the infinitives to be in the perfect form. If we choose the first in order to emphasise that we really wanted to do it at that moment, then the event itself is automatically past - it doesn't need specific grammatical marking. We therefore avoid marking the second as well because of what I've just said about English not liking to express meaning grammatically twice, and leave it other as a base form infinitive. On the other hand, if our regret is present, then it becomes obligatory to put the second infinitive into the perfect.

Despite the actual difference in meaning though, I suspect that the two forms are generally used interchangeably. It's relatively unusual that you wanted to do something at the time, but are now glad you didn't - or vice versa. So unless we are consciously trying to express that precise meaning, I suspect either could pop out.