It is my understanding, which is often flawed, is that once an assignment is made to a specific appraiser and that appraiser submits a report your client is responsible under FHA requirements to pay for the appraisal.
The way to insure that you're assigned the case number is to request the FHA Case Assignment sheets showing that you are assigned to that case number. Until that is done do not proceed with the assignment. A lot of clients who play the "value we need for pay" game will use many excuses to delay your assignment until they are sure the appraised value is what they need. Excuses will generally be "We've never had to do that with our other appraisers" or "I don't know what you're talking about. What assignment sheets?". And they will go to extraordinary lengths to delay your assignment. So, just tell them that until the assignment sheets are received that you can not proceed or only conduct COD business, or both.