One place I can think of that is going to let in underage people is a place called Los Socios [sp?] on Powell between Sutter and Post. It is a mexican restaurant that offers Karaoke but they have private room and I’ve no info about their music selection.
There is also a Karaoke place on Lombard near Webster called Silver Cloud, you might want to check and see if they have a private room.
Like it or not, J-town may be your best bet.
Per M.L.’s "answer" there is only one Karaoke place [The Mint[ anywhere near the Castro and it is a straight up bar with no private room.