Battle Royal Tournaments are not about racing, they're about attacking other racers. No lap times to worry about. The Tournament consists of 5 race tracks. The position that you finish on each track will award you points in the Tournament total. (10 for 1st, 8 for 2nd, etc.) To win the Tournament you do not have to win every battle, you just need to have the highest combined point total at the end of all 5. If you get 1st place in three races and second in the other two, you'll win.
Here is the key to Battle Royal: the shield. As you go between cones, your meter on the right side of the screen fills up. As your meter fills, you get access to different gadgets and weapons. You only need to use the first one, the shield. To cycle through available weapons/gadgets use the shoulder buttons (LB or RB).
Drive straight at another racer and activate your shield (X button). This counts as you hitting them. Hit a few racers and you'll get credit for all of them. Follow someone you just hit and wait until they stop spinning, then hit them again.
I got through all 15 Battle Royals (50cc, 100cc, 150cc, 5 tracks each) and the closest any AI racer got to me was 2 hits.