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$render_controller->view() invokes hook_ENTITY_TYPE_view() and this would be a better place to add custom values to the entity view.

Although probably not good practice you could get your approach working by cloning the entity before modifying the field values:

$entity = clone $entity;

In any case you need to adjust caching so that you store for each variant a different cache item in the render cache. For example by adding cache keys to the build array before returning it:

$build = $render_controller->view($entity, $view_mode);
if (isset($entity->foo)) {
  $build['#cache']['keys'][] = $entity->foo;
}
return $build;