The BES Client is running in LocalSystem context, and as such generally will not have access to network shares.
It’s possible to make it work, but exceedingly difficult. You will need all of the BES Client computers to have access to the share. This means the computers need to be in a group that have “Access this computer from the Network” right assigned (via Group Policy or Local Security Policy on the server hosting the share), as well as Share Permissions and NTFS rights to the file and path. The client will also need direct access to the network share through routing & firewall rules, as the downloads will not be distributed through the BES Server/Relay model.
A better option would be to host the file on a path accessible via HTTP/HTTPS, which can be one of your own internal servers (possibly the same server where the files are hosted now). Anonymous HTTP/HTTPS access would be needed, but you could restrict access to the IP address of the BES Server.
Perhaps you could use the Archive and Upload Manager tasks to get files hosted on the BES Server itself, but it sounds like your admin is trying pretty hard to prevent that.