Problem is the front. With a 15" rim you can't really get enough backspace to run any kind of decent width. Maybe if you could get a 245/55/15 on a 15x8 with a 5" backspace to work up front, but you'd need exactly the right rim to make that backspace work without hitting the suspension. Usually around 4.75" you start running into problems with suspension clearance and that won't clear 245's to the fender. I suppose a 235/55/15 would work better as that could be put on a 15x7", but that's only a 25.2" tire. And then if you go 235/60/15 you get a 26.1" tall tire, which is pushing it because of turning clearance on the lower front corner of the wheel opening with the backspaces you'd get. Maybe a 235/60/15 on a 15x7 with 4.5" of backspace, probably give you enough room that any mods to the lower front corner would be minor. But now you've still only got a 235 up front.
As for the tires, yeah, there are some track tires available in 15" that will work, but not even all of the autoX and road course classes allow them. Most of the popular classes that are fairly easy for the beginner/layperson to do, like CAM or even Optima, limit you to a 200 treadwear tire. So, it has to be a street tire, not just technically DOT legal. Most of the track tires are rated 100 or below for treadwear.
This is why 17's are so nice. 17x8's, 5.25" backspace, run 245/45/17's all the way around. Throw in a set of 1.03" torsion bars, front and rear sway bars and a set of Bilstein's and you're a corner carving mo-chine with a pretty limited amount of modifications, all reversible and none really all that noticeable (other than the 17's). Honestly, with the right tire height even the 17's aren't that noticeable, I've had to tell people I have 17's on my Challenger. And it's not just about how hard you corner on the street, better tire compounds mean better stopping distances and better crash avoidance, so when that Prius cuts you off while you're cruising slowly to Dairy Queen you don't take it out because your hockey pucks locked up and sent you sliding into the back of it.