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There seem to be two ways to solve this problem.
One is for the searches to be ordered on behalf of, and paid by, the client (that may involve taking the search money up front, paying it into client account, and paying the search fees out of client account).
The other is to recover the VAT paid to the search service provider and charge the client the search fee (excluding VAT) and then adding VAT onto the disbursement in the bill. It is how a number of other VATable disbursements are dealt with.