As someone who drives a lot in Europe...own car plus rentals, I would be very wary of cross border travelling and insurance implications. I would definitely consider renting with one of the majors and possibly paying 100 % insurance options. You need to call whoever you're thinking of and give them your itinerary for a bespoke quote.....you will be politely be told to go away by some as ,trust me, driving in some areas leaves a lot to be desired. Any parking is a gamble, whether that be break ins, particularly high value cars, or bumps or scrapes caused by unknown persons. You're talking of ancient towns and cities certainly not designed for today's cars.....narrow streets with locals having no regard to other road users or local traffic laws. Not all of course but you will be lucky to escape scot free on your proposed tour. I would rethink...you're on holiday, not a motoring marathon, and Europe will still be here next year. Speed cameras are rife, and in France speed detectors are illegal, my phone with Co Pilot on board was confiscated a couple of years back....i got it back...eventually.
And while Holiday Autos.com and others are fine for local rentals and are fine if taken at face value, but don't forget, you get what you pay for and no more. (Just see what happens if you break down on a Sunday, and does your rental actually carry a spare wheel ?) You need to ensure that all marks and dents, wheels,(alloy scrapes), tyres are scrupulously noted on the documentation and take photos all round, particularly on return if the rental office is closed. I might sound paranoid, but trust me , after 40+ years on European roads you get to learn what to do and what not to do.....it might well not be your fault in any accident , but guilty until proved otherwise.
Have a great time !!