I can't find much information about the ESC that's in your car. In general, LiPo has nothing to do with brushed/brushless, but it has everything to do with the ESC. You can definitely run a LiPo battery with a brushed motor if you have a LiPo-compatible ESC. So, you may ask, what makes an ESC LiPo compatible? Couple of things, but mainly having a LiPo voltage cut-off, so that the ESC doesn't let you fully drain the LiPo battery, and being able to handle LiPo voltages (7.4 volts for 2S, 11.1 for 3S).

So, step one for you is to figure out if your ESC can handle a LiPo. If it can, great, if not, you'll need a new ESC that can.

A brushed motor is OK, but it's pretty inefficient and harder to maintain. So, even a cheap brushless motor would be good for drifting. However, don't go crazy on kV ratings, you actually want something with more "torque" for drifting, as far as I understand it. So.. sensorless motors with lower kV ratings (such as 2400kV, etc.), or sensored motors in the 17.5 - 13.5 turn range, I'd guess.

Which brings up the topic of sensored vs sensorless. I'd say go sensored, as that gives you more precise throttle control, but it also means you'll need an ESC that can handle sensored brushless motors. So, your costs start adding up.

Not sure if any of this helps, but try to search for and read about the above topics, it should give you a starting point.