Ladies and gentlemen, I bring you
AWESOMENESS. Here is 3-3 beaten using the Kuribo's shoe to fall and run faster in water. FractalFusion suggested this to me ages ago but I dismissed it straight away (shows how good my estimates are, although I didn't know about the faster running speed).
For the numbers: W4 HB's (to get the shoe): 830-840 frames
Toad house (for a music box): 785 frames
This means that it would basically be better to just use a music box on the world 4 HBs,
except that now with the added help of 3-3 (and maybe other stages as well), it IS faster to fight one of the HBs for the shoe: the shoe saves
over 430 frames in 3-3.
Note that a music box should still be used for the second W4 HB because you only need one shoe. The difference is only about 45-60 frames though.
EDIT: Also, I timed whether beating a W1 HB and using the star in 3-2 is beneficial or not. Beating a W1 HB instead of using a music box sacrifices something like 285 frames. The star only speeds 3-2 up by 160 or so frames, so it's not worth it. Use music boxes on both W1 HBs and do 3-2 without a star.
EDIT#2: Getting a shoe inside a level (for example 4-3) wastes about 60-70 frames (unless some good fast strategy I don't know about can be used to get it more quickly) and since getting the shoe from the HBs wastes only something like 45-60 frames, getting a shoe inside a level doesn't seem to be a good choice.