I've done a bit of research on the Kenda Kaiser tyres the dealer fitted to your car. The reports on various forums are not good but are consistent in saying there appears to be some level of grip in the dry but almost none in damp or wet conditions. The very low price suggests they may not be up to much and the 350Z is very sensitive to tyres.
The quote below is typical of what I have read.
"The nastiest tyres I have ever had the misfortune of driving on. With a directional tread, and acceptable dry grip, they lure you into a false sense of security. However, at the first glimpse of damp, they lose the little traction they had and you may as well be driving on bald tyres. Uncomfortable, noisy, no feedback, and annoyingly hardwearing. A budget tyre to avoid at all costs."
I do agree with the others on here that it could all be down to the tyres. If the dealer agrees to put Bridgestones on there it would be good if you could get 050a rather than 040.
Good luck.