It some was ten years ago when we were waiting for the ferry at Patras and got talking to a Greek-Canadian family who had rented a car in Athens and were taking it to Corfu on the ferry. It's news to me that the car rental companies don't want you taking their cars on the ferries. Maybe that family had a cousin in the business, or maybe the rules have changed.
Anyway, feeling badly about misleading you, Roelle, I did some checking and found that you can take rental cars on ferries, but you have to have written permission from the company. http://www.dolphin-hellas.gr/CarRental/car-rentals.htm I guess you would want to check the cost of taking the car on the ferry versus the price of the bus.
The bus all the way from Athens to Corfu would undoubtedly be the cheapest way to go (about 30 euros, I think.) From what I've heard, the trip is a lot more bearable than it used to be now that the buses are nonsmoking.
But personally I hate buses and love ferryboats, so I try to avoid the Athens-Patras bus ride not to speak of taking a bus all the way to Igoumentisa! (and I must say, to drive that highway would be scary.) Usually we fly to Germany and on to Corfu on Condor. But we have also flown to Rome, taken the train to Brindisi and the ferry to Corfu. Unless one wants to visit Athens, it seems counterproductive to fly there and then backtrack to Corfu. But of course it all depends on what prices one can find from one's place of departure.