Sadly,it's one of those things that is probably illegal,but never gets to court.
Why?
Well, as you will realise if you dispute the sum the Letting Agent will probably serve a notice of eviction on you, when you come to renew again ,or will not renew the tenancy now.
It's called moral blackmail,and the law never sees it.
When I was younger there was a thing called "Key MOney",the agent in effect sold the tenancy to the person wishing to pay the highest "Key Money".
This was made illegal, but surprise surprise was still going on until fairly recently.
It survived because if you didn't pay the "Key Money" someone else did,and got the tenancy.
You COULD take it to court,but the agents have you over a barrel(as you know)and as bednobs so rightly says,they will fish out(from some musty drawer) their terms and conditions.
I do feel for you,but I can't see a way out.
I hope someone Legal oa AB might suggest something more positive.