Funny - I'm a Roadster owner in Phoenix and bought a used Model S from an owner in Seattle and had it transported to Phoenix.
Anyway, in my case, the seller left his valid tags on the car but he was not supposed to that. He also had a lien on the title, so I had to wait on that. In the meantime, I took my purchase paperwork/bill of sale to the DMV with proof of insurance and they issued me a 30-day temporary tag. When the signed titled came, I took that to the DMV and got my permanent AZ tags.
However, since you don't live in AZ, they won't issue you a temporary registration. If you can talk the seller into leaving his valid tags on the car and you show your valid insurance, notarized bill of sale, and signed titled you should be fine in case you get pulled over. It's not ideal to dive with that title around, but what can you do?
I'd research the requirements to get temporary tags in Washington state and find out if it's possible to obtain that before you fly out. Then, I guess, the question becomes how do you register car that you don't own yet?
Can you fly out to Phoenix, go to a bank with the seller, initiate a wire transfer, get the title, get a temporary registration online, print out the registration in Phoenix, and then drive home?