The way I am reading the rules, you don't qualify on any of that trip... it is only "International" or "Transcontinental" (with other restrictions) first that get access. (Of course, I am sure someone will correct me if I am wrong!)
As per the Admiral Club policies
here:
*International is defined as Europe, Asia, Central and South America, and Mexico City
*Transcontinental flights are non-stop flights between the following cities: Boston / Los Angeles New York (JFK) / Los Angeles
Boston / San Diego New York (JFK) / San Diego
Boston / Seattle New York (JFK) / Seattle
Boston / San Francisco New York (JFK) / San Francisco
Newark / Los Angeles Los Angeles / Miami
Fort Lauderdale / Los Angeles Miami / San Francisco
Washington Dulles / Los Angeles
Do you possibly have an American Express Platinum card? If you do, it will gain you access, otherwise, I would suggest a 1-day pass.