templating output of post blocks (or any output from shortcode-gen.php)

You can use one of the built in filters to customize the block any way that you’d like in your theme. On line #353 there is:

apply_filters( "mtphr_{$type}_post_block", $block, $excerpt, $args );

So, for posts you’d add the following filter:

function my_custom_block( $block, $excerpt, $args ) {
  return 'My modified block content';
}
add_filter( 'mtphr_post_post_block', 'my_custom_block', 10, 3 );

Because the post is created by a query that is overriding the normal query you can use:
the_id()
the_title()
the_permalink()
etc…