Run! Carnivore's outside in sheep's clothing (6)
I believe the answer is:
fleece
'sheep's clothing' is the definition.
'run carnivore's outside' is the wordplay.
'run' becomes 'flee' (I've seen this before**).
'outside' suggests removing the centre.
'carnivore' with its centre taken out is 'ce'.
'flee'+'ce'='FLEECE'
'in' is the link.
(Other definitions for fleece that I've seen before include "Con" , "Charge way over the odds" , "A sheep's woolly coat" , "A target for Jason" , "coat is needed" .)