You cannot insist that it goes to anyone. Fact is, he is a lawyer and he is entitled to act for his client.
Larke v Nugus is the Court of Appeal case from 1979 where the Court of Appeal said that in cases where Wills are disputed it was good practice for the Will draftsman to provide a complete copy of his file and a Larke v Nugus statement which details how the Will was drawn up, who was present, whether the deceased's capacity was in doubt, how instructions were given - basically the full process. There is more here:-
http://www.lawsociety.org.uk/advice/practice-notes/disputed-wills/
I would not refuse to send them copies. (If in fact you have the files - if you dont, you should make a joint approach to the firm holding them).
You say that his client has "no legal claim" on the estate - does the client benefit under an earlier WIll or intestacy? Who is this person?