I jumped into this thread a little late so I want to speak from my perspective which maybe somewhat subjective.
If you would want to track the car, it's probably best not to deal with the headaches of a force induction. The 370z is designed to be a NA from the drawing board if you want to make it a force induction and expect it to last at the tracks, you're talking about almost swapping or redesigning majority of the mechanical components.
Not that it's not possible, but to Cossie's point (in which I strongly agree), with that money, you're better off upgrading to another car with that money from the get go.
I am aware that there are a few people that have track their TT/SC 370z before, but the only ones that claim they don't have issues have either not pushed their cars hard enough, or long enough under stress. I track my car heavily and I'm already finding myself replacing/upgrading parts as a NA car. The 370z engines tend to run pretty hot, I'm afraid even after upgrading the radiator, vented hood, race oil cooler, with a NA if you're pushing it, you're already going to be at the high end temp of where you want to be. With a FI car, you might not be able to run your car hard throughout a proper 20 min session all day long.
When a car is designed with a FI right out of the box, the Design Engineers have done hard test at the tracks with that setup to ensure it works properly. From that these series of test, they could add necessary cooling components, beef up the thickness of shafts if broken, make stronger components, add air ducts, oil cooler, etc which are easier to do at the design stage.
I've tracked a Ferrari 458, McLaren MP4-12C, Lamborghini Aventador, Nissan GT-R at the track and even super cars such as these overheats after 5 mins of constant abuse. Esp the McLaren MP4-12C & the Lamborghini Aventador. Under the 110 degree weather, I did 5 laps and had to pull in for a cool down. That is why they have track versions of them that has better cooling ducts, etc which allows it to handle the abuse longer.
Another example would be the 450WHP
Crawford's Turbo Subaru BRZ. Sure they guy claim they could easily make a lot of HP from this engine, but they practically had to rebuilt the whole engine to ensure the engine could handle that kind of HP. Also, they broke a few half shafts in the process of building this car. So long story short, I say it's not worth it if you were to track your car unless you are ready to put a lot of money and time into re-designing the car. And by then, you can probably already be able to buy a like new used Porsche. (Which to OP's pt, won't be a bigger car but with the higher HP you wanted).
In the end the question is how much tracking are you going to be doing? I started off with 4 times a year and now finding myself doing 6 times a year and more as years went on. The more exposure and experience I gained, the harder I pushed and found more weaker links in the 370z. Another thing I can validate too is hardly do I find people who mod their cars that were originally NA to a FI have spent enough time to drive it well. I've seen ton of people that has FI cars which can't seem to put that power down to their advantage. You're better off spending that money on seat time or a car that was designed with more HP.
But if you're just driving it on the streets/freeway as a DD or just casual/spirited weekend mountain driving; if you probably built your FI 370z and takes good care of the proper maintenance, you probably can do just fine even if you push hard on it every now and then.
Hope this was helpful.