In addition to the user rights you have assigned, the NT service user has to be a member of the local administrators group in order ofr the Exchange integration to work.

When running manually, CRD actually connects using the local MAPI profile. The NT service cannot use this (if the pc is logged off) so its security requirements are somewhat more stringent as it bypasses the profile and connects directly to the exchange server.

Thanks, but the service account does also have local admin rights.

The error means that the CRD service doesn't have access rights to the Outbox of the account on the exchange server. If it works OK whilst trhe user is a domain admin, but doesn't work as a local admin, then clearly some restrictive security policy on your system, so you will need to get your sysadmin to look at policies and ajust them accordingly.