Regardless of what your client calls the search, you deserve to be paid for the work you are doing.  It sounds to me like they are asking for more than a typical current owner would include and I'd charge extra.  I agree with J.H., you are doing your part by disclosing exactly what you searched - the only issue left is whether you want to include the extra work in your basic current owner fee.  You might want to consider sending the client a fee sheet indicating when services you provide for your current owner fee and what additional services you will be willing to provide for additional compensation.

Best,
Robert A. Franco
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