Even though Lunala is an evolved form, it's still a legendary. It is the first ever legendary, alonside Solgaleo, to evolve. But like all other legendaries, it still can't breed.
If you want a Cosmog, take your Lunala to the altar of the moon at day time in game (night time irl) and go through the portal. Then you will be in a reverse world. Fly to the lake of the moon (which was called the lake of the sun before), go to the platform in the lake, and you will get a Cosmog. Then go back to the altar and you can return to your normal world.
If the daycare person says "but they don't seem to like playing together, though..." it means the two Pokemon you have deposited can't breed for whatever reason.