hook_field_extra_fields on node types stored in variable
hook_field_extra_fields is invoked by FieldInfo::getBundleExtraFields. This method caches the extra fields info in the database.

cache_set("field_info:bundle_extra:$entity_type:$bundle", $info, 'cache_field');


So you want to empty cache when the affected bundles changes. That is, add a custom submit callback for mymodule_admin_settings() and check what bundles changes over the previous settings.

Tip to Empty the cache:

$cid = "field_info:bundle_extra:$entity_type:$bundle";
cache_clear_all($cid, 'cache_field', true);


You may prefer to empty the cache for all bundles. In this case, $cid is the common prefix for all bundles:

$cid = "field_info:bundle_extra:$entity_type";