Great car!
I own the same model, it is called a Daihatsu Cuore over here in The Netherlands. Mine is a 2005 model, very basic without power steering, electric windows and AC. But weight is therefore even less, only 695 kg. And CO2 output is 109 g/km. I bought it new with 12 inch wheels with 145/80/12 tires, which were handling probably even worse than your 13" wheels. Recently I replaced them with 14 inch wheels with 155/65/14 tires, which is a big improvement in handling! Fuel consumption seems to have improved also, but I haven't driven enough yet to be sure about that.
Other modifications I have made so far are a grill block, scan gauge, removed the passenger side mirror and replaced the outside antenna for an interior one. I am running with a tire pressure of 3.0 bar (45 psi). Next on my list is installing a kill switch, have you already completed yours? If you do, I would like to know how you did it and what the result is!