Using a drush alias with Elysia Cron - alias permission missing
You could try drush @main --user=1 elysia-cron to force the active user to be the site admin when elysia-cron runs; however, all of the cron methods should be API calls, and therefore should not care who the active user is.  Note also that the error message that you quote above simply means that part of the Drupal bootstrap is failing for you; this is not a permissions issue (because Drush does not need permission to log on as a user -- it just does it), but is indicative of some other problem.

Run drush status with your cwd set to your site, and run it again vi drush @main status from a different working directory, and compare the output, esp. vis-a-vis the Site URI line.  Also run in --debug mode both times, and look for any other differences in the output.  If you have set root and uri in your alias, it pretty much should run the same both ways.