Happily consider feeding lady at fault (8)

I believe the answer is:

amusedly

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'happily' is the definition.
Both the answer and definition are adverbs.
Perhaps you can see a link between them that I can't see?

'consider feeding lady at fault' is the wordplay.
'consider' becomes 'muse' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'feeding' indicates putting letters inside.
'at fault' indicates an anagram.
'lady' anagrammed gives 'adly'.
'muse' inserted inside 'adly' is 'AMUSEDLY'.

Can you help me to learn more?

I've seen this clue in the Sydney Morning Herald.

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