Why? In the end, the borrower will get his hands on a copy anyhow if the OP does that.
There is no need for the OP to be an a**hole just for the sake of being an a**hole.
The OP already caused the borrower to contact the lender by his initial response to thew borrower and now the OP needs to to deliver a copy of the appraisal to the borrower as the borrower has caused the lender to address the OP's original objection to providing a copy.
I agree.
If any appraiser does not want to be in the position to provide a copy of the appraisal report to a borrower (or other 3rd party) then they should not say "I cannot do so without permission from my client".
Rich Heyn, in this forum, provided some excellent advice on this matter which I've incorporated into my reports. In sum, a statement that says
The named client is the intended user for this appraisal report. I am precluded by regulation, contractual agreement, and by my company policy to discuss assignment results or provide copies of the report to any other party. Any such requests for information should be made directly to the lender.
I bolded the above because Rich makes the case (as evident in this thread) that if the appraiser simply says, "I cannot talk with you/share the report with you unless my client gives its permission" then,
if the client gives the permission, it paints the appraiser into a corner. If the appraiser cites it is his/her firm's company policy not to discuss, then everyone has been put on notice that regardless of permission or not, the appraiser will not (or does not have to) provide anything to anyone other than his/her original client.
Of course, the appraiser can always make an exception to that rule if he/she so desires (assuming permission to release the appraiser from maintaining confidentiality is given).
So, Rich's advice (which I jump on) is to add such a statement to a report and refer any 3rd party to it to avoid the situation the OP is facing.
Now, if I were the OP, I'd provide the borrower the report, and include the referenced statement in my reports going forward.