Crossword Genius

A dog biting Athenian character that's pushing it (1,3,4)

Ross

I believe the answer is:

a bit much

I'm a little stuck... Click here to teach me more about this clue!

'pushing it' is the definition.
I can't judge whether this defines the answer.

'a dog biting athenian character' is the wordplay.
I cannot quite understand how this works, but
'a' is within the answer.
'character' could be 'mu' (Greek letter) and 'mu' is located in the answer.
'dog' could be 'bitch' (I've seen this before) and 'bitch' is located in the leftover letters.
This accounts for all the letters.
This may be the basis of the clue (or it may be nonsense).

'that's' is the link.

Can you help me to learn more?

(Other definitions for a bit much that I've seen before include "Too much to put up with, unreasonable" , "unacceptable" , "out of order" , "Butch ami may be excessive, beyond the limit" , "Somewhat unreasonable or OTT" .)

I've seen this clue in The Times.
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