What is the correct way to attach an arbitrary field from a different content type to an existing form?
I considered using field_attach_form(), but I can't wrap my head around what the $entity argument should contain.


It's the entity for which you want to show the form elements to edit its fields. For example, if you want have a reference to a node in $node, and you want to add the form elements to edit its fields, you add the following line to the code.

field_attach_form('node', $node, $form, $form_state, entity_language('node', $node));


The validation handler for the form should then call either one of these functions.


entity_form_field_validate()
field_attach_form_validate()


The first function builds the entity object from its $form_state parameter; the second function requires the entity object as second parameter.

Keep in mind that, despite the fact the UI allows you to add fields to a content type (e.g. add a Portrait image field to a Biography content type, but not to a Blog content type), for Drupal the entity to which the fields are attached is a node, which means you need a node object. passing as entity a string containing the machine name of the content type doesn't produce the expected result.