Diction simply refers to the style of one's writing or speaking. No matter how you speak or write, you are using some form of diction. So, to say that "pig-in-a-wig" is an example of diction is much too general.
Metonymy is a figure of speech in which one word or phrase is substituted for another with which it is closely associated. For example, if a news reporter says, "The White House announced today a new anti-terrorism program," what she really means is that the President, or one of his aides, made the announcement. The White House itself is just a house and cannot make announcements. "Pig-in-a-wig" is obviously not a metonymy.
So which figure of speech is "pig-in-a-wig"?
It could be considered an oxymoron: a figure of speech in which contradictions are placed side by side.
If you say that someone is like a pig-in-a-wig, then you have used the phrase as part of a simile.
If you say that someone is a p.i.a.w, then you have used it as part of a metaphor.
No matter how you use the phrase, let's not forget that it rhymes and that it is a poetic foot (rhythmic phrase) of four syllables whose first syllable is stressed.