You should get access. You won't need a pass, just go to the club in SEA and present your boarding pass and ID.
TRANSCONTINENTAL FIRST CLASS, BUSINESS ELITE OR FULL Y-CLASS PASSENGER
Customers traveling on a nonstop Delta First Class, BusinessElite or full Y-class ticket between New York-JFK and Denver, Las Vegas, Los Angeles, Portland, Phoenix, San Diego, Seattle and San Francisco.
http://www.delta.com/content/www/en_...-of-terms.html