I have been playing furniture games with the cat camper in my town. How long does it take before you have to do charades or rock paper scissors to make her move in?
It usually doesn't take too much time to convince a camper to move into the town, provided that you have enough room.
After the camper asks you about the town and perhaps whether or not it'd be nice to live there, any game that he or she asks you to play has the chance to be a game that determines whether or not that camper moves in.
For example, I recently found Maple camping in town, and the very first game she asked to play was one that involved guessing directions, saying that she would move in as part of winning the game.
To speed up the process, you can simply refuse any game the camper offers that isn't related to moving in, and repeat if you happen to fail one of these games. Since the camper hasn't moved in yet, none of these actions will offend the camper, and you are free to do this as often as you need to until the camper is convinced.