Don't skip six days because ANY villager could move away without warning.
Fast forward two days, find the villager that wants to move (could be any villager, not the one you want) and if it's the one you want to move, say bye. If it's one you want to keep, make them stay. Then either fast forward to their moving date (if you said bye) or go back two days to the original date. Then repeat.
You might not get the villager you want to move to ping straight away, just keep trying until they ping.