If output overrides aren't powerful enough to do what you want - there is another option: custom templates. This isn't really much different than the output overrides. However, there is one key difference.
You aren't limited to the file names of the existing templates. You can name your template filename anything you like.
Output Overrides (Review)
As described in the Output Overrides article, Joomla has an amazing system to allow a site administrator to modify the OUTPUT of the core without modifying the CODE of the core. By copying the plugin templates into JPATH_ROOT/plugins/content/apaapi/tmpl/ - you can override the template outputs provided by the plugin.
Custom Templates
I foresaw a scenario where a site owner might want more than the 3 options I provided. They might want 10, 20, 100 templates. The sky is the limit, but I suggest you come up with a naming convention to keep track of them all!
Filename Usage
As described in the Output Overrides article, the override paths don't exist on a fresh install. If you create them - Joomla will use the output override instead of the built-in template. That path is: JPATH_ROOT/templates/YOUR_TEMPLATE/html/plg_content_apaapi/
If you create an entirely new filename, let's call it "awesome.php" and place it in that directory, you can now specify the new template in your content tag like this:
{amazon asin=XXX template=awesome}
