This specimen has been submitted to the Grammar Lab:

The following turns up frequently:

I think we would have loved to have done even more.

You could say

. . .we would love to have done even more. or

. . .would have loved doing even more.

or . . . would have loved to do even more.

but I'm certain that "would have loved to have done" is wrong. It's like doubling the past and throwing in the conditional for lagniappe, which suggests the kind of time-travel found only in science-fiction.

Do you know the name of this fault?

I could sure utilize, as they say, your help.

The specimen sentence smudges the sequence of tenses in English. The perfect tenses — with an auxiliary verb attached to the past participle — work to establish distinctions in the time of actions or states of being.

We would love to have done even more expresses a feeling in the present about a past action.

We would have loved to do more expresses a feeling in the past about a past action.

We would have loved to have done even more is a jumble. The present perfect — have plus a past participle — indicates a past action compared with a present time. This sentence has two conditional past perfects without establishing a clear relationship to the present. The alternatively worded sentences clearly establish a relationship in time.

There is a useful entry on verb tenses at Purdue University’s OWL (Online Writing Lab).