Airlines such as Ryanair in Europe do this.
One of the regulations that airlines in europe are required to adhere to is that they're not allowed to advertise up-front prices that come with additional non-optional fees. So the ticket price advertised up front has to include all mandatory fees and taxes. Optional fees (hand luggage now being the latest, priority boarding, checked luggage) can be extra.
Payment processor fees cannot be extra. At least one needs to be free. So in reality, it's not necessarily the cheapest processor that's free, but it's the least popular one that's free. That way, for all the common ones, you can now charge a fee and tick the regulatory box, while still screwing your customers out of more money.
E.g. Visa Electron is free, Visa Debit/Credit, Mastercard debit/credit and Amex all come with fees to use.