Storing current page WP list table HTML data into a PHP variable

Ah!

function callback($buffer) {
  // modify buffer here, and then return the updated code return $buffer;
}

function buffer_start() {
  ob_start("callback");
}

function buffer_end() {
  ob_end_flush();
}

add_action('wp_head', 'buffer_start');
add_action('wp_footer', 'buffer_end');