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The definition of "segment" for the pass is an enplanement. So you do need two for a connection. As for the correct fare class issue the pass requires booking into a specific fare bucket on the flights. They may actually not be available. You can try checking seatcounter.com to see. I forget exactly which bucket they book in to for KE but a quick search here should turn that info up.
Also, my understanding is that the pass has to be issued in conjunction with an intercontinental flight into the region. So it would not be available if you lived there and are just planning a regional trip.
Your troubles in issuing the ticket sound familiar. They don't issue many because they are rarely a good deal so the agents never learn about them.