This is the relevant RCW page: http://apps.leg.wa.gov/rcw/default.aspx?cite=46.61.055

It doesn't specifically mention this situation, just that "Vehicle operators turning right or left shall stop to allow other vehicles lawfully within the intersection control area to complete their movements.", which to me doesn't necessarily mean you can't enter the intersection.

However, probably the reason why people think this is illegal is that it is explicitly illegal in California and it is of course a well-accepted fact that all the bad drivers in the Seattle area are, in fact, Californians.

The 3000 mile/3 month oil change is a myth perpetuated by Jiffy Lube. Almost no cars made in the last 20 years has had a 3000 mile change interval, even for "severe" service (my '86 Honda has a 7,500 mile one, for example). 5,000 to 7,500 and 6 month to 12 month are more common. Your owner's manual is the absolute authority on this-- follow it, not the oil change place or random yahoos on the internet.

Fight it in person.

Additional Options:
Cry a little, and make it clear you made your first ever driving mistake and don't have any way to pay for it, let alone have it mar your record. Mention the new driving job you just interviewed for that you'll lose if the boss finds out about the ticket.

Mr Cop, by the way, was telling you in hardly-uncertain subtext, that stopping is actually important, and while he nailed you for the stop sign, he doesn't approve of the hefty fine/punishment. (His job's law enforcement, not law making.)

Well you insert it, and then (at a later time) remove it quickly.