Custom Content Entity type - allow fields to be managed
Alright, after lots of research I've come up with three different solutions to the issue.

Solution 1 (not so great):

On the text formats and editors configuration page (/admin/config/content/formats), the top format in the list that the user has permission to will be set as the text format automatically when saving the field (entity). If you only have a text format or two, this will work, but it is easily breakable, so it's not a great solution.

Pros - easy.

Cons - easy to break if/when someone adds a text format or re-orders them. Hard to configure. Not stable at all.

Solution 2:

In your form defintion for the entity (or a _form_alter hook for the form), you can set the format as follows. Note that in this case, my field is keyed as 'message:

public function buildForm(array $form, FormStateInterface $form_state) {
  $form = parent::buildForm($form, $form_state);

  foreach(\Drupal\Core\Render\Element::children($form['message']['widget']) as $element_index) {
    $form['message']['widget'][$element_index]['#format'] = TEXT_FORMAT;
  }

  return $form;
}


Note: Replace TEXT_FORMAT with the key of the text format you want to use.

Pros - can be overridden as part of the form by other developers

Cons - can be overridden as part of the form by other developers (if you want to ensure that the text format is always the same and cannot be changed)

Solution 3:

In your entity definition, first add this function (again, my field is keyed as 'message'):

public function setMessageFormat($format) {
  $this-&gt;get('message')-&gt;format = $format;

  return $this;
}


Then you can add a preSave() method to the entity definition to set the format:

public function preSave(EntityStorageInterface $storage_controller) {
  parent::preSave($storage_controller);

  $this-&gt;setMessageFormat(TEXT_FORMAT);
}


Note: Replace TEXT_FORMAT with the key of the text format you want to use.

Pros - Ensures your text format is always what you want it to be for that field on the entity

Cons - cannot be changed by anyone else if you want this to be configurable.