Unnerve Republican who’s got a nasty bite? (7)
I believe the answer is:
rattler
'who's got a nasty bite?' is the definition.
(rattlesnake)
'unnerve republican' is the wordplay.
'unnerve' becomes 'rattle' (I've seen this before).
'republican' becomes 'r' (abbreviation).
'rattle'+'r'='RATTLER'
(Other definitions for rattler that I've seen before include "Nickname for a pit viper in the US" , "excellent specimen" , "Nickname for US snake with horny segments in its tail" , "I'm shaken!" , "Poisonous American" .)