Another system is using an amiibo/getting lucky and Nook Miles Ticket having a brand new villager camp on your island. It's a little cheaty though. When you eventually persuade them (after hours of playing that frickin card game) and they request to kick out someone, keep reloading the game (go to the dash without continuing the dialogue and shut the game without saving) till they suggest the one you need gone. If I'm not making sense, you will find guides online that describe this better than me
Not sure if you understand so that I hope this suggestion helps, but the basic process for moving individuals out is that every 2 weeks, a random villager will have the cloud bubble above their mind and will ask to move out. If you find the cloud over someone that you want to keep's mind, don't speak to them! The bubble won't be on them the next day but it will not restart the two week cycle. 1-3 days after another random villager will be selected to have the move out bubble. Just keep leaving them thinking (do not speak to them at all that day) till it randomly selects the one you would like to move out.
The new remarks they provide you after a very long absence are sometimes pretty thoughtful, too.
It's always okay! They are always surprised to see you afterwards however long for the very first conversation, then it is back to Animal Crossing Bells For Sale how it was before you went away.They do not become angry or threaten to leave or anything, iirc they just mention it's been x months and that they missed seeing you around or anything similar.
