I think it is helpful to have an overview of the protocol for getting to a 1004P in the first place. FNMA must first have the neighborhood data. Second they must have a recent (within 6 years?) appraisal with which they run a PIW. If they have the data but don't have a recent appraisal then they give the option of ordering an inspection. The inspection data is fed to FNMA so I assume they may know who the inspector is or maybe they have just agreed with the vendor that the vendor is responsible for the data.
FNMA takes the inspection data and feeds it into their AVM. If FNMA's AVM does not hit the number (range with confidence intervals) needed to waive the appraisal then the option for the 1004P desktop is permitted (It is an option to the borrower). Then, and only then, does the appraiser see the order.
This is the current protocol. It may very well be that if FNMA is successful with the 1004P (it is currently only permitted for high quality borrowers) then we could probably see that getting pushed further down the line WRT risk (just my opinion).
As an aside I have seen six of these 1004Ps for actual loan purposes, and only one of them hit the number needed to make the loan (a renovated house). My takeaway from this is that FNMA is only doing these on homogeneous housing with lots of data (cookie-cutters), and it is unlikely that any type of appraisal would hit the number (except for example the renovated house).
So in the end the borrower is stuck with an appraisal that can not be sent anywhere else, and once Realtors figure out that the 1004P is not a 1004 will not be accepted for price re-negotiation purposes.