The relationship api does require the administer civicrm permissions.

We have a site where some users have permissions to create memberships for contacts. Those users have the right permissions but they lack the permission administer civicrm and that is on purpose because they are not administrators. On some membership types civicrm does an api call to relationship get count but this fails because relationship get count api requires the civicrm administer civicrm permission. This is set by default by the api and as it seems this should be a bit lower than administer.