Question: How do we end up with three vowels, εου, at the end of these genitives? (Or morae if we are going to say ου is a vowel.)
The last vowel, I venture to guess, comes from -ος, the genitive singular suffix for third declension nouns. I know that it sometimes combines with a stem ending in a vowel and undergoes a change, as in γένους from γένεος.
But to have three vowels together, we would need a stem ending in two vowels or somehow get an extra vowel in there between stem and suffix.
According to Wiktionary, the etymology of Περικλῆς is:
περῐ- (peri-, “very”) + κλέος (kléos, “fame”) + -ής
Could it be that between εο (from κλέος) and η (from -ής), we somehow ended up with a stem ending in two vowels?
Relevant too seems the case of of Σωκράτης. Its stem is Σωκρατε(s) (according to Mastronarde's Introduction to Attic Greek). The (not necessarily etymological) operation that takes you from word to stem is: (1) remove -ης, yielding Σωκρατ, and (2) add ε(s), yielding Σωκρατε(s). The same operation on Περικλέης gives you Περικλεε! Or at least there is some sort consistency here.
I would note that the question above on the genitive recurs for
the dative and the accusative as well, e.g. Περικλεῖ and Περικλέᾱ, where we get three morae at the end, if not vowels. Compare ἀσπίδι and ἀσπίδα; Σωκράτει (Σωκράτεϊ) and Σωκράτη (Σωκράτεα). (Note: I struck Περικλεῖ because it does not have three morae.)