The lone exception to that rule is Grumble Volcano, where you use Baby Mario + Bit Bike due to the handling increase. Here is a list of all the courses and the character/bike used.
Luigi Circuit: Funky Kong + Spear
Moo Moo Meadows: Funky Kong + Flame Runner
Mushroom Gorge: Daisy + Mach Bike
Toad's Factory: Funky Kong + Flame Runner
Mario Circuit: Funky Kong + Spear
Coconut Mall: Funky Kong + Flame Runner
DK Summit: Funky Kong + Flame Runner
Wario's Gold Mine: Luigi* + Mach Bike
Daisy Circuit: Funky Kong + Flame Runner
Koopa Cape: Funky Kong + Flame Runner
Maple Treeway: Funky Kong + Flame Runner
Grumble Volcano: Baby Mario + Bit Bike
Dry Dry Ruins: Funky Kong + Flame Runner
Moonview Highway: Funky Kong + Spear**
Bowser's Castle: Daisy + Mach Bike
Rainbow Road: Funky Kong + Flame Runner
Retro Courses:
GCN Peach Beach: Funky Kong + Spear
DS Yoshi Falls: Funky Kong + Spear
SNES Ghost Valley 2: Funky Kong + Flame Runner
N64 Mario Raceway: Funky Kong + Flame Runner
N64 Sherbet Land: Toadette + Magikruiser
GBA Shy Guy Beach: Toadette + Magikruiser
DS Delfino Square: Funky Kong + Flame Runner
GCN Waluigi Stadium: Funky Kong + Flame Runner
DS Desert Hills: Funky Kong + Spear
GBA Bowser Castle 3: Daisy + Mach Bike
N64 DK's Jungle Parkway: Daisy + Mach Bike
GCN Mario Circuit: Funky Kong + Mach Bike
SNES Mario Circuit 3: Daisy + Mach Bike
DS Peach Gardens: Funky Kong + Spear
GCN DK Mountain: Daisy + Mach Bike
N64 Bowser's Castle Daisy + Mach Bike
* Daisy is faster than Luigi, but the WR uses him.
** Moonview Highway is done using the automatic drift mode.
Hope this helps