Getting a second set of wheels and going with a 200tw on one set and something above 300tw on the other and only using the 200tw on track or while autocrossing will save you money in the long run and cut down on excitement levels when it rains.
Asking a set of tires on a daily driver to handle street and track duty is the worst sort of compromise. On a second weekend car sure, but on a daily, no, not at all. Additional wheels are needed. (and that's before we get into having a conversation about tracking an only vehicle).
If not going with a second set of wheels, I'd look at conti ECS and will second the suggestion of an aggressive alignment using the adjustable camber aspects built into the front of the TT. I'd strongly consider playing with alignment settings and adding an adjustable rear sway bar before doing suspension, being a TT you'll be working on wanting to dial out understeer to make it neutral, but having an idea of how much plow and push you are working against will give you a baseline with regards to what you want to do with regards to spring rates down the road. I always strongly recommend tracking a car in nearly stock form for a year before making changes (unless the model is known for having oil starvation issues that can be avoided via a change in plan design) for a few reasons. You'll learn the car, track and driving on track, if you decide that it's not for you, or the chassis isn't and that you'd prefer a different starting point, you're not out the amount invested in modifications. You'll also learn where to focus modification efforts, what about the car you want to change or correct. This most often results in brake, suspension and tire related adjustments.
If you have a budget to spend before going on track next spring, I would get a nice set of pagid or feredo pads, good rotor blanks and really good brake fluid. I've never heard someone come back from their first track day, or any track day, proclaiming that they had too much brake. Bias set not optimally, yes, but never too much brake. And in the rare time you hear an argument of too much brake, it's really an example of not even tire.
After brakes I'd save for another set of wheels even if tires won't be on them by spring for track use.
Then enjoy a few sessions, come back talk about plow and push, whether and when tires got greasy and after how long, how your pedal feel was and how long until you got fade. Then we can start on sway bars, possible brake ducting, tires, changes to alignment from your current setup and possibly even potential spring rates.