I suppose it stands to reason that if DL can restrict access to third party operated lounges to J pax only, so can the other Skyteam carriers. I do somewhat understand the frustration however as KLM does not document this fact anywhere on their website.
On a side note, I was flying on KE DPS-ICN in coach last month and the KE agent gave myself and spouse lounge invitations for the third-party "Premier" lounge. It seemed a bit odd as Skyteam member Garuda Indonesia has a lounge right next door. Why wouldn't KE just direct me to use the Garuda lounge?