I would advise staying on the side from which you will be leaving. There are plenty of places in the town of Foz, as well as some more resort-y places on the road between the town and the park, the same road that passes the airport.

It really isn't all that hard to get around and over the border to the other side of the park. I did it by straightforward local public buses and didn't feel inconvenienced in any way (although I do rent a car for other, less accessible locations in Brazil, and to get around in, for example, the Yucatan and Turkey). Not adverse to renting, in other words; just didn't feel it necessary or any major advantage, in this area.

Whatever you decide, enjoy.