You can do the Prime Instant (the free, I haven't paid for a video yet on Amazon) through the
browser, not an app. Browsers like Dolphin and Opera are pretty good about going to the "real" sites, so you don't have to worry about getting mobile-only stuff. It's probably a Fire-only app that's giving you the compatibility error, or Amazon is being just vague enough on their wording of what the streaming service is to make you think you have to buy a Fire to get it.
Update: Liliputing reports that the Amazon Video app on the
Fire can be loaded onto other Android devices.