Without time travel or amiibo cards, patience is the only way to get rid of any villager. Hitting them with a net and ignoring them doesn't do anything to make them leave. Fifteen days after a villager has moved out, the option for another villager to move opens back up. If a villager has a thought bubble over their head, they may want to move. When you talk to them, they suggest moving, and you can agree or convince them to stay. If you tell them no, a villager will take another five days to ask to move. The villager that wants to leave is mostly random, though villager who's just asked to move out, had their birthday recently, or just had their house moved won't ask. So you can either wait and hope Rodney asks to leave, time travel until he asks to go, or get an amiibo card and request him to move.