Hi all, A little late to the conversation. While I can't speak to registering a custom bike, I don't believe it would be much different to my experience buying a bike off amazon Japan. These bikes are not preregistered and should be registered with your local city office. However that was easily done for me. I took it to a bike retailer. In this case it was Cainz. This shop is not a bike shop, but more like a home depot or Canadian Tire (depending on where you used to hail from. Any place that will do a basic bicycle tune up will do. You drop it off. They check to make sure everything works. The gears. The brakes. etc. You pay a small fee. Upon request they will register the bicycle and provide the registration tracking sticker for your bicycle. Boom. Bob's your uncle!
It was fairly easy and I'm fairly confident even without Japanese language ability, your bike, your I.D. and google translate will get the bike registered at pretty much any bicycle resale/tune up shop.
Good luck all!