Anonymous users can't access to image style field
The purpose of a field configured in a view mode is to render the output.

To process field values in code use the entity object. You'll find this in a block template in this variable:

$variables['elements']['content']['#block_content']


So in TWIG you can try to access the field value like this:

{%
  set style = elements.content['#block_content'].field_image_style.target_id
%}