Shikoku may seem small, but driving across it can actually be rather time consuming, particularly if not using the toll highways. The same holds true for driving anywhere in Japan actually. And in the back country it can really take a good amount of time to cover what seems like short distances (using google map directions can give a rough time estimate, but this won't factor getting stuck behind a slow moving vehicle and the innumerable stop lights placed seemingly everywhere). So it'd be best to not try to cram too much into your schedule, and be aware of the daylight time (it gets dark about 5:30 or 6pm in October)
Yes, the Shimanto is the last un-dammed river, though most rivers don't have massive Hoover dam-style installations, just low-rise sluice dams to manage flooding.
And the pools and waterfalls of the Niyodo River area are indeed beautiful, but not all allow swimming (but some do). Check for info here
https://niyodoblue.jp/en/
One really nice campsite out in Iya Valley is at the Oku Iya Niju Kazurabashi (the double vine bridges). I stayed here a few years ago and it was really cool since you camp right next to the vine bridges and swimming in the river is possible. Its pretty far up in the valley though, but this place is much nicer than the other Kazurabashi vine bridge which is lower in the valley and much more congested. But on the way you can also stop at some pretty spots including the freaky Scarecrow village which was close by. Here’s an article about that campground 1:
https://miyoshi-city.jp/2021/09/9319/ 2:
https://miyoshi-city.jp/2021/09/9336/ 3:
https://miyoshi-city.jp/2021/09/9350/ (in part 3 it explains a forest bathing hike you can do from here)
And there are lots of great (and not so great...) onsen everywhere in Shikoku. It's worth checking around before you go, especially with closing days. Many people say Dogo onsen in Matsuyama city is a must-do, but to be honest, its a bit over-rated. Let's just say, after going once many years ago, I have no compelling need to go again. I prefer the quiet mountain-based onsens which have a big nature element and aren’t bubbling with tons of tourists. In Iya, the Hotel Kazurabashi onsen (which is near the lower valley vine bridge) is definitely fantastic. And over between the Niyodo and Shimanto Rivers is the Soudayama Onsen, which was also super tranquil.