See? Needless confusion.
1. yes, the third wasn't mentioned, but there was the Hozuki guy (2nd Mizukage) and Yagura (4th), then Mei (5th) so there has to be a third Mizukage.
2. when Yagura was shown to be under the control of Tobi, Tobi and Kisame were almost at their current age and Yagura was a kid, one of the youngest kage in history... and when he died, according to Naruto, he was younger than Naruto himself (15 or 16 at best). It is essentially impossibile that Yagura was already a kage when Obito became Madara's apprentice, I'd even question if he was born at all. Yagura clearly wasn't a Kage during the third war.
Now, if you keep attention in the "Obito+Zetsu VS Kirigakure shinobi" fight, you'll see that the village was already known as "Bloody Mist" by that time and its ninja even bragged about it being labeled as such. All of that fight does OBVIOUSLY take place before Yagura's (Tobi's) reign.
It does make sense, then, that Yagura hosted Isobu AFTER Rin. Any other explanation makes no sense, in fact.
3. the only way to go around this is saying that Tobi was Madara at the time, and not the "current" Tobi (Obito), and Kisame looked somewhat older than he was for some reason, and when Obito took the mask away Kisame mistook him for Madara because of yet some other weird reason, such as Obito disguising himself as Madara with genjutsu or so. Which is your theory as I understand. Point here is that the timeline is not screwed, it becomes screwed if your theory is taken as true.
4. Last thing, Naruto had contact with all the previous jinchuriki in his spiritual experience. There it is confirmed that the last jincuriki of Isobu was Yagura, which therefore comes after Rin.