Amazon downloads on the Roamio will not play on the Tivo mini if the show is selected from the Tivo mini menu. If you try to play it on the mini you will receive a "Copyright message".
Here is the work around that works for me -
I have the Tivo App downloaded to my IPad.
Open the app.
In the drop down menu that lists your Tivo devices - Select the Mini that you want to watch a program on.
Now, go to My shows in the Tivo App.
Select the Show that you downloaded from Amazon.
Press Watch Show.
The show will now play on the Tivo mini that you selected.
I have confirmed that this will work on the Tivo App for Iphone and the Ipad.