Anyone who has been with you for a while and will move for a price reduction is not worth calling a customer. See this as a good thing, as it's allowed you to cut some customers from your round that you don't want.
Move on. You don't want reluctant customers. You want people who are happy to pay you for a decent service. So keep looking for the right people. Leaflet/canvass/advertise until you have a round of people who value your service.
Do your job correctly and price becomes, within reason, irrelevant,
As for 'having a word' with your competitor, why would you want to do that? You should be encouraging him to take your price conscious customers. Get rid of them as soon as you can. He'll be losing them as soon as someone cheaper comes along. What's he going to do then? He's got nowhere to go but lower prices.
Vin