After a lot of digging around I did figure out a way to do this.
In the App's app.php file, you can force a script to be loaded if the plugin is enabled:
$api->addScript('script_name'); // without .js
In that script jQuery can be used to add the elements to the page where you need them.
This was a good solution for me since I only needed to add a single button to the login page. I can see this being a bad idea if you want to make vast modifications. At that point you might as well just create a separate page that you have full control over.