At JFK, the DL lounges are airside in T2 and T4. You had all DL operated flights then if you flew CLT to JFK followed by JFK nonstop to LIS, NOT KLM (although it might have been a KLM marketed codeshare, but this seems unlikely).
My LIS tickets were booked through KLM and were codeshare with Delta. I don't remember who operated which legs, but I'm aware of what airlines I booked tickets through. Do people not know who they fly with? That seems odd, but thanks for your input.
I also know it was a pain in the a$$ because each leg coming home was a hassle for some reason. Almost missed my flight ATL-CLT because they couldn't "release" or "unhook" the leg initially and it took 3 different Delta employees to get me on the plane.