You can run track days without super sticky rubber too you know
In fact, if relatively new, it might be advised by some to NOT show up on super sticky rubber so as to aid in your learning process.
You can run on just about any size wheel (15", 16" 17 etc - how large is the budget?)
On my SC a few years back I ran Toyo RA1. 225-50-16 front and 245-45-16 rear on 8" front and 9" rear fuchs. Same tires also used on 7" front and 8" rear fuchs. I ran the toyos as street, AX and DE tires. I beat folks on hoosiers.. Some beat me.. in the end, the driver is more important than the tires (at the local club level it seems)
While yes, you want AX tires to heat up quickly, if you were serious about AX, you'd run Hoosiers or V710 Khumos, no? The false issue about being "quick" at DE is that, there are no timers.. no prizes.. its just ego..
If you are running well on the car currently with the Dunlops, I don't know that you would need to change "mid stream" as it were.
The biggest limiting issues will be on TIRE availability. There are fewer and fewer 16" sizes that work well on the 911. Jumping to 17" wheels would help this a little bit (think 235 front, 255 rear)