Grammatical construction of "to have happen"

In a sentence like the following:

What a wonderful thing to have happen on your birthday!

what is the grammatical form of to have happen? I know the sentence is grammatical, but I can't identify how. It isn't a perfect infinitive, as in the following:

What a wonderful place to have visited on your birthday!

because to have happened would be incorrect in the original sentence. I thought it might just be the infinitive of a phrasal verb have happen, as one can in fact say something like:

I have happened to visit her in Atlanta quite recently.

But on reflection, the latter construction just appears to be the present perfect of happen. So I discarded that idea.

I searched online, but 'have happen' grammar and 'have happen' definition didn't yield anything useful.

Start with "Something wonderful happened to me", copy "me/I" at the beginning as a new subject and insert "have" as the verb for that subject after the Aux complex. This gets you "I had something wonderful happen to me". Now optionally omit "to me", giving "I had something wonderful happen". Your example is an exclamation corresponding to this.

I don't know that this construction has a customary name, but it is quite common. I have seen it compared to the Japanese "adversative passive" construction. It topicalizes someone/something affected by an event or condition. In the case of the affected person or thing possessing something inalienably (like a body part), the possessor is ordinarily deleted from its original position:

A mole is on his nose. (= There is a mole on his nose.)  
He has a mole (*be) on his nose.  
He has a mole on the nose.  

Evidently an original verb "be" is lost.