Mirror number of unread mails in title bar
You can obtain the number of unread items with the following AppleScript:

tell application "Mail" to get the count of messages of mailbox "INBOX" of account "Me"


As you mentioned you could write a menu bar item, you could use this AppleScript for it.

Alternatively, you can do this with Mail Unread Menu.
However, I'm not sure if it works with Mavericks.