The new Fit is slightly heavier than the Yaris, but still handles better... the old Fit was hands-down a better car.
There are multiple issues with the Yaris as an autocrosser... you need to modify the suspension, because the stock set is biased towards massive understeer, and allows way too much body roll to utilize the tires properly. The electric power steering is very, very, very numb... and just a few minutes of hard driving will overheat the motor and leave you without assist... (which isn't exactly a bad thing... feels better that way)... but the stock radiator is also undersized, and that saps power, too.
You can make it stick. You can make it quick. You can probably make it competitive. The Mazda2, however, will be better. It's less powerful, but it's lighter and faster at low speeds, and it has sharper handling. The Fit should also be better, too. Both generations have better engines, and the rear suspension is better (toe in under hard cornering, and a rear sway-bar as standard).