Just for those who have the mod installed can benefit from it 😄

There is no way for the mod server-side only. Maybe you meant client-side? For now the mod is required on both sides, but maybe there could be an option to make the mod still work if the mod was not detected on the server (in which case, all cats would be always loafing as there would be no reliable way to save this information)

I thought about this: https://modrinth.com/mod/let-sleeping-dogs-lie

Client side would be fine too.

I thought that it might still be possible for players to benefit from the synchronization if they have installed the mod. But allowing vanilla clients to connect.

Oh, I see what you mean. I will look into it and see if it's possible without redoing many things (I use Minecraft's built in entity data synchronization system, which probably wouldn't allow mismatches between client and server)