Just do what brad siad,....he got it right
Years ago I tried the compromise of good road tyres on the track and it is just frustrating.
Much better option is 2 sets of wheels that you swap before the track day. Buy some cheapies in whatever you want for road use with OK rubber on them, and then put semi slicks on the 17x9 (assuming they are good wheels). THat is a great size for reasonably priced semi slicks around 245/45/17 up to 265/40/17.
The compromise of trying to use 1 tyre for 2 totally different uses will just frustrate you for both uses....and it is more expensive over time because you will rip up road tyres on the track