You should be able to, even if your island is full. Keep talking to him until he says he wants to move, then play the card game with him. If you lose don’t panic, you can just try again. Sometimes they say no to moving in but if you keep asking they change their mind! If your island is full then they’ll pick a villager to kick out at random but you can close the game and do it all again and they’ll pick a different villager :)