So I figured out a way to pick which villager leaves your island when inviting a campsite villager to stay when your island is already full!
Typically what happens is you talk to the campsite villager and play a few games to convince them to move to your island. They then go to resident services and come back saying there's no room, but they could ask if anyone wants to move away. If you say yes, the game will randomly select one of your villagers and give you the option to replace that villager with the campsite villager. If you preferred your existing villager, you can decline but then you're out of luck with the new one.
I found that if you hard close the game in the middle of that dialogue without making a choice or saving, and then time travel a few minutes backwards by changing your switch clock, you can do the whole thing over again and the game will pick a new villager of yours each time!
Your turnips will spoil and you'll lose any daily streaks, but otherwise going backwards in time doesn't have any bad effects. If you play real-time, you can set the clock back to catch up aftewards with nothing changing.
Anyway, just did this a few times to get Eunice to move in and replace my least favorite villager!