This is a fairly balanced view of the different services. I don’t trust Amazon after they disabled and deleted content off the device. Just one time of being disrespected is enough.
Netflix vs. Amazon Prime Instant Streaming. It seems clear from this review that Netflix is still the better value. Although Amazon Prime is only roughly a dollar less, it has far less titles. Nice try Amazon! The video quality does seem to be higher however.
If you want to do this your best case is to make a case at Apple for it. If you have Applecare then you can legitimately ask how you can accomplish this. If I were to do this I wouldn’t expect to be able to save all the data. I would expect at the very least that 3rd party software would miss things like smart playlists, and metadata.