there's nothing wrong with the oem wheel and tire size.
It might help getting something lighter to reduce unsprung weight. You're probably not going to be able to run significantly wider rubber or wheels without spending a lot.
In my first event ran stock suspension, 16x8 wheels, 205/50 200tw dunlop z3 tires (not significantly wider that what you have), EBC orange pads up front, not that different from what you have, ss lines endless rf-650 fluid, boomba deflectors. This was in 90f weather.
oh also had upgraded radiator, fmic, fmic hoses and oil cooler. my guess is oil cooler and fmic hoses were probably not necessary.
If you don't want to spend much maybe try just upgrading the tires and call it a day. Some people think upgraded radiator and fmic not necessary but I think it's worth it.
If you do eventually replace the pads, highly recommend hawk dtc-30 pads up front for street and track, if you don't mind a tad bit of squealing on the street.
Also highly recommend setting front camber to more than -2. maybe -2.5. Def play with tire pressures.