Absolutely take the ballcap. We each take one on every trip.
For sun protection, I have several Sloggers hats. They fold up (mine has been stuffed in a suitcase and a tote bag multiple times, and it bounces back beautifully without creasing), or can be packed flat....have a chin strap (great for when out on the water or where it might be windy). Mine are about 10 years (or older), so the colors I have (black with just a smidge of white in the braid (favorite for travels) and also a dark tan one (hardly ever wear) do not appear to still be available, but if black/white would work for you, here is one of the current model ones:
https://www.sloggers.com/Sloggers-Women-s-Braided-Hat-UPF-50-p/442bw.htm
Amazon also carries Sloggers. I originally purchased mine at a garden center attached to a large True Value Hardware store.
But, definitely take a hat of some sort, and a comfortable one is best.
Forget all the don't wear hats because you will look like a tourist advice, and follow the protect-yourself-from-the-sun. Pink with a daisy will likely make the French smile :)