It is possible to send a transcript for a conversation from your inbox in Fuse IM.
When you are viewing a conversation, the right hand visitor information panel has a number of options. Near to the bottom you will see "Email me a transcript" and "Email <user> and transcript".
"Email me a transcript" will email the transcript only to you, whereas "Email <user> a transcript" will send a transcript to the user if the conversation has an email address.
Clicking either of the options will ask you confirm first.