Why don't you just get a copy of the death cert from the GRO yourself?
Researching at the GRO isn't free. There are fees to look at each of the record books. The indexes are part of the books which have the detailed death records. There isn't a separate index. The records aren't computerised. If another researcher has a particular book it could take hours for them to finish with it and there may be others in the queue likewise waiting for that particular book.