I agree. You definitely want to run lower pressures at the track. Hopefully someone here has a suggestion for you as to what pressures exactly.
Honestly, unless you plan on running near-race pace, it is not rocket science. The track is just a road, right? So, for a lap, ride at a conservative, almost normal road pace (in the corners and braking zones). Second lap step it up a little. If all you are trying to do is maintain a novice or low intermediate pace, then a couple laps on street tires is going to get them hot enough to support that.
I use only street tires at the track (albeit a bit sportier than what you have) and no warmers. I find that I can pretty much get on it after one warmup lap, and after two laps I can put in lap times that are at the faster end of the intermediate group.. My cold pressures are 30F 28R (Dunlop Q2s) but the 2CTs may need something quite different.
Yes it's a bit tougher when it is cold, but these are street tires right? They are supposed to work at those temperatures. Give'em a couple laps and they will be fine. Now, if we are talking like 5-degree temperatures...then I can't help ya. Never tried it, likely never will! Take it easy on the throttle - your rear tire will tell you if it is warm enough or not!