I would never drive in Italy, but that's me.
But answering the OP's question about where to stay while keeping their rental car, my top suggestion would be on the Aventine. It's outside the ZTL and it's a beautiful, calm and quite pricey residential area. There are a few B&Bs and I know that the Hotel Aventino on Via di S. Domenico has free parking. But it's based on availability so that might be too much of a gamble. However, I did a quick scan on airbnb and saw a couple of apartments that included parking. From the Aventine you could actually walk to where you'd want to go. Or, hop on the B Metro at Circo Massimo, or there are many bus lines that serves that area.
The other suggestion would be in Prati, which is also outside the ZTL. Tons of lodgings there because of the Vatican's allure. So it's a matter of finding who offers parking.
RoninRome who used to post here has a page on ZTLs in Rome:
http://www.roninrome.com/%20transportation/ztls-in-rome