How do I (in general) track down what happens on form submit?
You mean updating the field instance settings programmatically, in an update hook, hook_update_N(). Do it as follows:

function YOURMODULE_update_N(&amp;$sandbox) {
  // First load the field instance.
  $instance = field_info_instance($entity_type, $field_name, $bundle);
  // Make the changes in the field instance configuration.
  $instance['widget']['settings']['apply_chosen'] = 1;
  // Save the field instance.
  field_update_instance($instance);
}