Moralising Republican, beastly on the outside (8)

I believe the answer is:

priggish

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'outside' is the definition.
Both the answer and definition are adjectives. Perhaps there's an association between them I don't understand?

I don't understand the rest of the clue.

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(Other definitions for priggish that I've seen before include "Self-righteously superior" , "Exaggeratedly proper" , "Strait-laced" , "Sanctimonious" , "smug" .)

I've seen this clue in The Times.

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