Getting a new appraisal for a cash out refi

[Len P.]

Hello everyone! I have just completed a rehab on a buy-and-hold, and am looking to do a cash out refi with a conventional 30-year loan. The appraisal came back at $110K and I am confident it should be closer to $120K. The bank has told me they cannot order a 2nd appraisal now because it will appear they are value shopping. They are bound for 120 days with the current appraisal.

If I cancel the current loan application, can I then do the loan process with a different bank in which they would order their own appraisal? Or would other banks also be bound by the initial appraisal.

Thanks!

Len