You are projecting and guessing what YOU would do if you were a lender, but that is not how most of them operate. They already know what the appraisal fee is for most normal res non-complex orders. It is already established with their own panel or with an AMC if they use an AMC. A lender does not have time to dick around with different appraisers over a few bucks on each deal they quote borrowers for.
The lenders have a close range of title and RE appraisal fees they work with on regular properties for the good faith estimate. If a borrower is threatening to go to another lender because of the total costs, a lender can shave $ off their points or fees. Whether they choose to do so or not can be deal-specific.
A high end or complex property that needs an appraisal quote is another matter. Though many lenders now also have a wide range of higher fees they charge for jumbo loans, up to a point where a custom quote is needed.
The entities with the incentive to dick around with appraisers over their appraiser portion of the appraisal fee for regular orders are the AMCs, because that is how they make their profit. Lovlely little leeches they are, sucking $ from the most poorly paid and hardest working on the RE food chain.
A lender makes their profit from the loan, not from keeping a split off an appraisal fee.