Entity Translation for custom entity
In my case the issue ended up being pretty simple to fix. I found that the function entity_translation_form_alter() was testing to see if my entity's edit form was suitable for alteration. My edit form failed the very first test because entity_translation_edit_form_info() was returning false.

entity_translation_edit_form_info() was returning false because my form didn't have $form_state[$entity_info['translation']['entity_translation']['edit form']] set and set to the entity I was trying to edit.

tl;dr: Make sure you set a value in your edit form's $form_state at the path you declare in the edit form element of the settings array for your entity returned by your implementation of hook_entity_info().