I beat him eventually with help from items (a double lightning if I remember correctly), but I have been unable to beat him in time attack mode, where there are no items. Also, (slight spoiler coming up)

After you hit Bowser 3 times, he rolls into a ball and you have to flip him out of the room while also keeping Mario out of the hole. It sucks.

Also, I am online playing Mario Kart pretty much every evening from 6.40 onwards. 090 255 381 818 if you want a match. This is me, throwing down the gauntlet!