I’ve never transferred a conventional appraisal, and think the issue in your case might lie as much in whether the NEW lender was willing to accept the prior one’s lender as much as whether first lender would agree to transfer. Lenders have to “warrant” to loans' end investors that appraisals were ordered in a manner compliant with guidelines, and for a new lender to do that on an appraisal they didn’t order is a relatively large risk.
Was your rate locked before the costs went up? Lenders must document a “changed circumstance” to alter fees on a loan once a Good Faith Estimate has been issued, they can’t just spontaneously decide to charge more. If they do, or even if they understate third party fees such as tax or title charges, recording, etc, the lender can be held responsible for the underdisclosed costs. There is some leeway for lenders, depending on where the cost arises from, but LENDER fees cannot be changed in any way after they’re disclosed unless credit score, loan to value ratio, or another significant item creates a “changed circumstance.” Would love to hear the explanation on what costs changed and why. Anyway, hope that helps.
Bottom line, the probability of a conventional appraisal being transferred is not good.