Why do you think Moon was more understanding to the Avarius family (and presumably the Monster King) than to Buff Frog? My theory is that Moon places a distinction between "civilized" monsters and, well, "savages," especially with Buff Frog's reputation and resume.
The way Moon knew where Lord and Lady Avarius were residing shows that it’s not the first time she’s engaged with them. Moon has her prejudices, but as with most prejudice, it’s usually dampened by constant positive interaction.
As fellow royals, the Avariuses would have been instrumental in maintaining the tenuous peace between the Mewmans and the Monsters. They would have had to agree with the lands they got. In the same way the Monsters rallied around Ludo in the first season, they would have been looked up to and listened to.
I don’t think it’s because the Avariuses are royals per se but being royals would have allowed them chances to interact with Moon. She would have known them and seen them as struggling in a similar way, as rulers over a kingdom. That dispels a lot of the negative preconceptions. In the same way when she finds something in common with Buff Frog, she gets along with him much better.
It’s the same way someone can have Minority Friends but still say that inequality doesn’t exist and Those Others are asking for too much. It’s a mental exercise in compartmentalisation, to me at least. Because Moon has never read like a character who hated the masses and loved only the elites. She’s perfectly willing to call out the nobility, as we’ve seen in Game of Flags, and Moon the Undaunted. She cares about her people and at times infantilises them (but that’s not very different from the way she patronises Star and River, so we can chalk it up as an aspect of her character).
I hope this helps!