You will [most probably] get your B/E boarding pass @ JFK when you check in for ATH.

I am fairly certain that that CRC staff @ SEA will admit you once you explain your situation:

"Hi. I understand an onward ticket overseas in B/E allows club access here in Seattle?"

"Yes, sir. That is correct. May I see that boarding pass please?"

"I'm stopping over in New York on the way to Athens, so I don't have it yet. Here is my itinerary."

At worst, they will ask you to wait while they look up your itinerary to verify that it includes a B/E seat ex-JFK later.

And, yes, there is a "special" B/E lounge @ JFK. The buffet stampede might amuse you.