I realize I'm resurrecting an old thread here, but I have a little bit of experience that might be relevant.
Years ago I registered a 250cc dirt bike as a street legal motorcycle.
I jumped through a few of the hoops people here have mentioned, went to the DMV, got told to go to the State Patrol, etc...
What I ended up doing was adding all the lights and street legal tires necessary to pass inspection, (made a lot of trips to Bent Bike, getting all that stuff up and running with oddball parts and no experience is harder than you might think). Once I had that done I took it to the WSP for inspection, they gave me a piece of paper saying that it was in order and then I went over to the DMV where I forked over about $200 to get a license plate for it.
The long and short of it was that it was a PITA, but doable. Had I known what I was getting into I would have just sold the bike and bought a street legal bike to begin with.
I imagine a similar thing could be done with a motorized bicycle.
First off, I would start by telling the DMV up front that you have a "homemade moped" that you need to get registered. Don't tell them it's a "bicycle" or even a "motorized bicycle". The DMV doesn't handle those. They handle "mopeds" and under Washington law that's what you've got.
Second off, it's got to be possible. Just think of all the choppers and homemade trailers that get built, registered and licensed. They're not all being built with fraudulent VINs. The real issue is whether it's worth the trouble.
IMO, the real devil in the details is likely to be the tires. I strongly suspect that in order to pass inspection you will need DOT approved tires. That means motorcycle tires, which also means motorcycle wheels and once you start going down that route you're talking about one big, HEAVY motorized bicycle. If it were me, I'd think about taking them back off again after the inspection.