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just buy a more portable bike bag, I keep banging on about this, but took travelling bike round the world with pipe lagging and cable ties in the ground effect tardis, which was easy enough to transport, and not too heavy when I wasn't stuffing it full of clothes as well..
http://www.groundeffect.co.nz/products/tardis-compact-bike-bagtook it on the bullet train with no fuss, just another piece of luggage, albeit slightly larger than everything else.
Also if you're going to be using the bullet trains you want to get the JR rail pass, which is valid for 7, 14, or 21 days, you can buy over here and then get it validated over in japan to use during your stay..
Yes, they go in luggage compartment so they need to be in bike bags.