Cervelo Soloist, Giant TCR Advanced, Co-Motion Americano Co-Pilot
I started (similar route, fully supported, transam W to E) in mid-June and finished on first of August. I would say starting in early June, maybe even late May would have been OK (only concern would be getting over Rockies in May) and ending up in September would be great.
I agree with bobframe, leave as early as the weather permits. The first hurdle is the North Cascades Highway, which is usually open sometime in May, but sometimes doesn't open until June. The second hurdle are the Rockies. As long as you are in the area, I strongly recommend going over Logan Pass in Glacier NP. Unfortunately, this usually opens late June/early July, so if you go that route, you will need to leave later. The websites for N Cascades NP and Glacier NP have information on road conditions, updated daily once they start plowing, so I would keep an eye on those as it gets closer to the summer.