Limesurvey Sync incompatibility

Interesting. Not sure why there would be a default importer with that url (with "edit" between feeds and mailhandler_nodes). I tried your suggestion and the steps matched what was in my system, but it did not cause the Import tab to show when I open a mailhandler source node. I did rename the Mailhandler Feed Importer as well as make clones of it that create different types of nodes. Not sure if that is of any relevance.
I've now deleted the "default" importer that showed up when I followed your steps. If I now re-open admin/structure/feeds/edit/mailhandler_nodes/settings, that blank importer comes back. Still no re-appearance of the Import tab.

Ok, found a fix! (but don't exactly understand what the issue was).

Seems that "Import" is being stepped on on the menu structure somewhere.

I edited feeds.module, hook_menu(), changing import to import1 in just one place:

  $items['node/%node/import1'] = array(
    'title' => 'Import',
    'page callback' => 'drupal_get_form',
    'page arguments' => array('feeds_import_tab_form', 1),
    'access callback' => 'feeds_access',
    'access arguments' => array('import', 1),
    'file' => 'feeds.pages.inc',
    'type' => MENU_LOCAL_TASK,
    'weight' => 10,
  );

and that did the trick
Clear caches to see the change

Not sure what is stepping on the Import tab, or why this was needed.

Ideas?

I believe limesurvey_sync module is the offender that is stepping on the Import menu.

In feeds.module there is:

function feeds_menu() {
  $items = array();
...
  $items['node/%node/import'] = array(
    'title' => 'Import',
    'page callback' => 'drupal_get_form',
    'page arguments' => array('feeds_import_tab_form', 1),
    'access callback' => 'feeds_access',
    'access arguments' => array('import', 1),
    'file' => 'feeds.pages.inc',
    'type' => MENU_LOCAL_TASK,
    'weight' => 10,
  );

and in ls_survey.module there is

function ls_survey_menu() {
  $items = array();
...
  $items['node/%node/import'] = array(
    'title' => 'Synchronization',
    'page callback' => 'drupal_get_form',
    'page arguments' => array('ls_survey_import', 1),
    'access callback' => 'ls_survey_import_access',
    'access arguments' => array(1),
    'type' => MENU_LOCAL_TASK,
    'weight' => 15,
    'file' => 'ls_survey.import.inc',
  );