Is "pig-in-a-wig" a diction or metonymy or another literary device?
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.