I always find the "Death Race" with Beastman to be the hardest.
Anyway, Battle Mode has the same 10 Races. Just you VS them in a random order.
Mr. EAD's Mission is pretty easy if you take the right path, then the left, after that, smooth sailing (providing you don't hit the wall hard). The best bet is to pratice that track in Time Trial Mode.
If your aiming to get a good time in Battle, your only option is to use "Dr. Clash", since that Mission gets replaced with a Mission with Guster and is only a one lap race on a different course. 
As for beating Dr Clash's Mission, which was one of the Fire Field levels (one of the easier ones if I remember correctly), go and pratice in Time Attack mode.
@customracer209: The Violence Missions are random as well as the opponents. The game gave a me a Death Race with him once on the last mission.
Failed because of him.