When bottling, is it wise to dunk the bottle capper in sanitizing solution or is that overkill?
Probably overkill. Assuming the caps are sanitized, the capper does not come into contact with anything that it could spoil or infect on the bottle. At that point in your process, bottles and caps will be so covered by sanitizer, I'd say youre safe.
If you want to get really technical, you touch the handles of the capper, after which you may touch the inside of a cap before placing it on the bottle or on the capper magnet. But it's overkill.