1) The yoke isn't rated the same, and you may damage the chassis or yoke.
2) The G07 13" tube neck has a different setup than the G07 19" tube neck. This means that the sockets may not be wired the same, and you may end up throwing too much voltage down the wrong pins, possibly damaging your tube or chassis.
3) The CBO is designed to send a signal large enough to fill a 19" tube. Sending that signal to a 13" tube may damage the tube, but would probably give you such a distorted picture that you wouldn't be able to determine if the problem was the chassis or the tube.