If it's an FHA loan, that's not going to work. The case number can only be assigned to one company. Even if it's not an FHA loan, you will run into issues doing this. In the time it takes you to search for 2 lenders, order and pay for 2 appraisals and worry about all of this, you could have found one completely competent and reliable lender to close your loan for you. Huge waste of your time and many others if you ask me.