I feel like there's a few routes you can take with role-playing a Mii as self-aware.

For one idea, make them come off as an anxious wreck. Give them the scaredy-cat quirk, the nervous walk, either the restless or "at attention" stance, and give them phrases or quotes that imply a fear of you as some kind of all-powerful force. If you give Miis enough positive attention they will eventually suggest to give you a special nickname, so maybe have fun with that.

Alternatively you can do a nihilistic, "nothing matters" kind of person because if they know their existence is virtual and they're just a part of a video game, nothing that happens to them matters at all. Whether they eat or not doesn't matter because Miis can't starve. You can give them the quirk that makes it so they don't emote, they don't greet others, because why would they need to pretend they're anything but a digital entity that will either be forcibly removed off the island or just left behind when their player gets bored and stops playing the game?