Dangerous enemy is a challenge to the beautician (4,8)
I believe the answer is:
ugly customer
'a challenge to the beautician' is the definition.
I can't judge whether this defines the answer.
'dangerous enemy' is the wordplay.
'dangerous' becomes 'ugly' (I've seen this before).
'enemy' becomes 'customer' (I can't explain this - if you can you should give a lot more credence to this answer).
'ugly'+'customer'='UGLY CUSTOMER'
'is' acts as a link.
Can you help me to learn more?
(Other definitions for ugly customer that I've seen before include "Dangerous antagonist" , "Hard case" , "viewed unfairly by shopkeeper?" , "Thug" .)