Well... I suspected that it could be the computer and tag problems and it's not the airline's fault. I have looked at the virgin australia's tag. It has got three stops/transit cities only on tag. For example, the tag can be printed "SYD/LAX/Detroit" - no more added as it runs out of space on the tag. I don't think that they will give you another tag on same bag.

I can't understand why the airlines staff cannot explain about the tag problems.

It would be better if you can pick your bags at LAX, then check-in again.