Can Settings API setting generate other settings?

I believe you should be able to use the update_option_{$option} hook:

do_action( "update_option_{$option}", $oldvalue, $_newvalue );

Something like:

add_action( 
  "update_option_youroptionname", 
  function ($oldvalue, $_newvalue) {
    // process your option value and update/insert options as needed
    // var_dump($oldvalue, $_newvalue);  // debug
    // wp_die(); // debug
  },
  1,2
);

Be sure to replace “youroptionname” with the appropriate value.