Day one: commission campsite
Day two: (if you paid it off on day one) celebration
Day three: guaranteed camper.
What you want to do is on the third day DO NOT LOAD UP ANY MAIN CHARACTERS. Instead select to create a new character and go through the sequence with Rover. Once you can move freely in the town, go check the campsite. If it's not someone you want, quit without saving and do this procedure again. If it IS someone you want, continue on through the motions of making a new character, set their house, blah blah blah then once you are able, save and quit then load up your mayor or whatever and go to the campsite, empty your pockets of anything important to you, bells, tools, items, etc. Pick up a couple of things like seashells or your native fruit to have in your pockets. Go inside the tent with the villager you want and spam talk to them a million times until they bring up moving to your town. They might say no at first, but you can still convince them. It might take a while, but keep talking to them and eventually they might make you play a game to determine if they move, if you lose it's not the end of the world and they'll take one of the items you have in your pockets (which is why you've dumped all important ones). Eventually they will agree to move to your town on their own or if you beat them in a game they will agree to move to your town. They'll introduce themselves and make some comment about how they are excited to become your new neighbor. Until this happens all other responses are rejections. You can also know they've officially said yes if you talk to them again and they talk about what life will be like living in the town or something along those lines and they'll start asking you to play other games with them and stop asking if they should move to your town (obviously because they've already decided)
Hope this helps!
Also when you do this make sure there are no ropes placed in your town and you have under 10 villagers. If you have ropes I don't think they will move in. I don't know for sure. But also don't panic the next day after you convince them to move in because it takes them two days to move in from campsite so
Day 3: convince whomever to move to your town
Day 4: no ropes
Day 5: Camper will place ropes for their house in your town.