Under the bar is the correct way and i doubt the tremolo system would be effective if you strung straight through.
If you strung it like a LP, SG you would'nt have any 'break angle' from the bridge to tailpiece. It could result in poor sustain, poor tuning stability, a useless tremolo system and at worst string jumping (out of the saddle slots).
Having said that it makes the string tension very flabby when you raise the tail piece on a LP or SG, its comfortable especially for lead playing. If you sank the cigar tube as low as possible you may just get away with it but given its a short scale (less tension anyway) i dont see any benefits.
