I was curious about this, so I just tried it today, and now I see why there isn't a current category for it. You unlock different car parts by playing mini-games, but on a clear save file, you have to play Story or Party mode to get those mini-games, since you have to play them at least once to have them appear in the free play mini-game mode. You can do Easy and Normal tournaments easily with the basic car that doesn't require any mini-games, but the difficulty ramp with Hard makes the basic car incredibly unlikely to win. Additionally, the maps where the fights happen are pretty random, so in addition to the odds being stacked against you in Hard, there's also strong RNG based on the map you get. A shame that my most nostalgic part of MP5 isn't a good speedrun fit, but it was worth a try!

I'm four years late, but I also was wondering this! I came to this game to look for the Battle Car specifically, and I'm surprised to find it missing.