I've been cycling villagers over the past three days, here's how I did it:

[CowKing]

Could you explain more? I've never been through this process before, is there dialogue that says which villager will be negotiated with?

Sure!
When you talk to a camper that isn't Amiibo and you convince them to move, they'll try to get in contact with Residence Services, but if your island is full, they'll come back to say the island is full, but if there was a villager that wanted to move, they could take their place. The camper will then ask if you know a villager who wants to move, if you chose 'yes', the camper will chose a random villager to negotiate with. If it's a villager you want to keep, immediately close the game as soon as they say the name, before they're done with their dialogue (The game doesn't save during dialogue, only after) and try again. It'll let you convince the camper to move in again and the next villager they try to negotiate with will be random. I was able to get Tammi to move out and get replace her with Ketchup with this method.

 

[ThePhantom]

Sure!
When you talk to a camper that isn't Amiibo and you convince them to move, they'll try to get in contact with Residence Services, but if your island is full, they'll come back to say the island is full, but if there was a villager that wanted to move, they could take their place. The camper will then ask if you know a villager who wants to move, if you chose 'yes', the camper will chose a random villager to negotiate with. If it's a villager you want to keep, immediately close the game as soon as they say the name, before they're done with their dialogue (The game doesn't save during dialogue, only after) and try again. It'll let you convince the camper to move in again and the next villager they try to negotiate with will be random. I was able to get Tammi to move out and get replace her with Ketchup with this method.

That's actually really clever! That makes the campground more useful than I thought. Thanks for explaining

 

[PikaPika19]

I have a question: I'm using this method to just move out one villager I need to move out. What do we do if a villager we want to keep gets the thought bubble? Is there a way to "reset" (for lack of a better word) who wants to move?

 

[Wolfycheeks]

Does this work when you haven't reached a 3 star rating yet?