Active perfect infinitive

Solution 1:

There is no present perfect infinitive.  Infinitives don't have tense.  They don't mark either the present or the past.

One way to show the difference is to use a verb that has a separate infinitive form.  The verb to be happens to have that property:

I was happy to be finished with everything early.

No present-tense form fits.  We don't use "to am finished" or "to is finished" or "to are finished".  No past tense form fits.  You won't find "to was finished" or "to were finished", either.  For to be, only "be" fits.

I was happy to have finished everything early.

In the same way, only "have" fits.  Sure, "have" is a simple present-tense form, but it is also the infinitive form.  Only the infinitive form fits.

It is reasonable to call this a perfect infinitive.  However, it is a non-finite phrase, and it is not reasonable to label it as either past or present.  It is a tenseless perfect phrase.  It works regardless of the tense of the clause:

I was happy to have finished.
I am happy to have finished.
I will be happy to have finished.

Solution 2:

[1] I was happy [to have finished everything early].

[2] *I was happy [to had finished everything early].

The perfect is a past tense that is marked by means of the the auxiliary "have", which is inflected for primary tense ("has" or "had"), as in for example, "Ed has been ill" / "Ed had been ill". These constructions thus have compound tense: 'present perfect' and 'past perfect' respectively.

In [1] "have" is in the plain form, so there is no primary tense, no compound tense, though it is still perfectly grammatical. It is usually known as the 'perfect infinitival' (or 'infinitival perfect').

But [2] is ungrammatical since "had" is a tensed form and thus cannot be used with the infinitival marker "to".