Lobbyist’s head in stew to compose catchphrase (7)

I believe the answer is:

tagline

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'catchphrase' is the definition.
(I've seen this before)

'lobbyist's head in stew' is the wordplay.
'lobbyist's head' becomes 'l' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'in' is an insertion indicator.
'stew' becomes 'tagine' (I've seen this before).
'l' inserted inside 'tagine' is 'TAGLINE'.

'to compose' acts as a link.
This may not be right. Some or all of it may belong to another bit of the clue.

Can you help me to learn more?

(Other definitions for tagline that I've seen before include "catchphrase" , "Slogan" .)

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

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