The BESAdmin (or the BESXAdmin, how you named it) needs Send As rights. If you have given the Send As rights in Exchange, and it has lost it again, make sure that BESAdmin is not member of the Domain Admins group.
Details on how to give the correct rights via Power Shell:
Q. How do you set the BESadmin permissions in Exchange 2007?
A. From the Exchange open the Exchange Power Shell and run the following two commands:
get-mailboxserver "<mail_server_name>" | add-exchangeadministrator "BESAdmin" -role ViewOnlyAdmin
get-mailboxserver "<mail_server_name>" | add-adpermission -user "BESAdmin" -accessrights ExtendedRight -extendedRights Send-As, Receive-As, ms-Exch-Store-Admin