I'm using Live App Testing to test In-App Purchases. Here's how I've found the process to work:
1) I create a new test in the developer portal.
2) Email invitations are sent to my testers immediately. Testers can then use the links to "buy" the app, but the app doesn't appear for download in the Amazon Underground app.
3) Wait about 24 hours.
4) Then users open the Amazon Underground app again and now they can download the app.
My suggestion is to not send the invitations until the app is actually available for download. That would save a lot of time and avoid the issue of users forgetting to go back 24 hours after the email to download the app.