It's irrelevant whether the car is within break in period or not. The car can drift on dry pavement. The cheapest tires you can find that fit your wheels will work fine. Just don't daily it with those same tires unless it's really nice and warm out.
For reference, I've drifted my car since day one. 14k miles, over a dozen sets of cheap tires, third set of brake shoes, and enough drifting to be on my third rear axle as well. No issues outside of the snapped rear axles because I'm doing something way out of what was intended with the vehicle. May look into aftermarket axles if this one goes at some point.