Dear Ioannis,
I've searched around for a solution to this, and figured out that a special userdb {} section in the protocol lmtp {} section, with driver = ldap etc etc so that i 'd be able to "show" the correct user to the lmtp might work, but strangely that userdb {} setting looks like being ignored by lmtp.
A very similar issue was discussed on IRC last Friday. userdb is indeed ignored on the director, as the different process are going through the special "director-userdb", which has nothing to do with the userdb you have define... The documentation (https://doc.dovecot.org/configuration_manual/authentication/user_databases_u...) was updated to make it cleaner.
The solution is to use passdb instead. Use basically the same configuration as the one you have for userdb but for passdb. For the ldap configuration itself replace user_* by pass_*.
I hope that helps, Vincent