Why are you wanting to go with the skinny tires? its a small difference but could lead to some odd handing issues if your don't change the fronts by the same ammount
Sometimes the electronics cant cope with different sizes and thow errors, also the 350Z isnt devoid of arch gap and putting narrower, lower profile tyres on wont help.
Mate, absolutely don't do it. The Zed likes standard tyre sizes and matched compounds. Mixing & matching will just send the TCS & DSC into meltdown and cause lots of issues at roundabouts in the cold, pissing rain.