Out of all the vehicles I've transferred titles over in WA, and some out of state, I've never had them ask for an engine number, there isn't even a place for it on the title? I've also had to have vehicles inspected by the state patrol and they have never looked at engine numbers.
Try taking the title to a different DMV, they are not all the same. Some are run by the state and some are private. Usually the private one's don't know their heads from their asses. They also charge you an extra fee and conveniently forget to mention it.
The good thing about them being idiots usually means you can sneak stuff pass them, like registering a dirtbike and getting a plate for it.