Trackdays don't have to be about going as fast as you can. They are great for working on skills such as looking through turns, holding a line, using reference points, practicing threshold braking, and learning throttle control.
The stock tires are adequate if you plan on running in the B group at a street pace or a bit higher. The danger is pure street tires aren't designed to run as hot as a track pace will get them. They will heat up faster than track tires, and the first two or three laps will be fine but if you push them hard (knee down through the turns and heavy trail braking) then you run the risk of overheating them and causing them to get greasy and slick.
The most important thing is that you have FRESH tires. Even tires designed for track use can have issues if they are old or have seen too many heat cycles.
If there is a shred of doubt in your mind, get new tires. The confidence they instill will help you enjoy your track time even more.