The reason new customers pay less is so that they can GET the customers. They take a "loss" on these customers to get them.

I'm fully behind the idea of asking for a discount. If they refuse, then you can either pay the price that every other "existing" customer pays or you can leave.

Some companies have retention departments that will offer you something (e.g., discount) if you try to cancel. Be aware, though, that you can't bluff. If they don't/won't offer you a discount at that point, be prepared to cancel.