Run lazy-builders of sub-keys in the render array
This seems like an anti-pattern. Normally you would configure and theme separate instances of the block for the fields you want to display in each region. 

If you need block fields in the page template, get the fields directly from the custom block content and put them in variables in page preprocess, see Programmatically accessing block content, or use the module Twig Tweak:

{{ drupal_field('field_example', 'block_content', 1) }}