I've never had to pay a fee for the inventory to be carried out. I've typically paid around £200 in "administration fees" at the start of a contract, but nothing at the end.
As others have said: double check your tenancy agreement. I also did some searching, and found
this site, which has an example similar to yours (labelled as The First Property). If you were charged an inventory check at the start of your tenancy, it is unfair to pay any fees at the end, and they cannot legally require you to pay (quote the Unfair Terms in Consumer Contracts Regulations 1999).
Even if you didn't pay at the start, it still doesn't seem fair-- given that the property has been sold!! I would definitely argue it. Let us know how you get on!!
In my experience, once you start quoting legal requirements and regulations at letting agents they tend to back down or make the necessary changes quickly.