You would need to switch to one of these if you want meaningful improvement. Frankly i wouldn't spend the $'s. Just enjoy being on track. It will take a lot of mods to make the car much faster, while still safe. Brakes are the weak link.
https://www.tirerack.com/tires/TireS...earDiameter=20
These will easily fit on the stock rims. i would go with the Mich. They are still very manageable on the street. The Toyo's would suck in adverse weather. Go big or go home. Any lesser tire and the incremental improvement will be underwhelming. You said it earlier, 5,400lb is a lot to overcome. That will overwhelm even race compounds. Surprisingly, or not so surprisingly, run flats are an ideal choice for the track. The stiff sidewall makes up for some of the 5,400. I've used both stock RFT and a comparable non-rft and prefer the rft on track and non-rft on the street. Since it sees more street, i use non-rft. Everything is a compromise.
I've been tracking cars for about 15 years. My current toy is a 1996 spec miata with a turbo. Cheap fun. Really cheap fun. My tow vehicle was a 2011 x5m, and is now a 2012 x6m. I've had one or the other X on about 5 different tracks in and around TX. Even bone stock it is no slouch. Surprises everyone.
video from a few years ago at Motorsport Ranch outside Houston