At least from my experience end last year, at KIX, there is only 1 choice for AF or KL passengers, which is a 3rd party lounge. I don't believe that there has been any changes to lounge usage - this 3rd party lounge has been used since years...it is also used by Air Calin passengers mainly.
During the morning bank, the lounge can get full due to all 3, KLM, AF and Air Calin flights departing around that hour. Lounge itself is OK-ish from both F&B and service point of view. Only possible issue would be finding a place to sit if it is crowded. It does have a rather good view of the tarmac.
Cheers!