Here's a possible route I thought up:
1) Complete the first floor.
2) Before doing any basement levels, go up past the blue wall (see trick:
"slip under the slab" and pound the pillars.
3) Continue through to Shifting Sand Lands. If possible, manipulate Mips 1 so you dive at him and miss, providing a little comedy while entering the level.
4) Get Mips 1 on the way to the Volcano (if possible to do so. Otherwise, get him on the way to Shifting Sand Lands earlier).
5) Get all the Volcano stars, causing Mips 2 to appear. Catch him as well on the way to Hazy Maze Cave.
6) Continue to the Hazy Maze Cave and get all the stars there.
7) Get the metal cap star last (and the cap itself, if any stars actually require it. Do they?).
8) Let the current take you down the waterfall to the waiting vanishing cap star and cap. Return via menu to lobby, then basement.
9) Get the the Dire Dire docks stars, and the Bowser star.
10) Instead of exiting via menu, super stairwell boost up the basement steps to get to the courtyard and Boo's Haunt.
11) Get the Boo's Haunt stars, exit to lobby, continue upstairs, get the stars there, finish the game.
Alternatively, if the metal cap star doesn't automatically kick you out, you could get the vanish cap first, skip Hazy Maze Cave completely, do the other stars, and then enter Hazy Maze Cave through the metal door after being dropped outside via Wing over the Rainbow. Get the metal cap star as the next-to-last star in the game, then exit to the lobby to go upstairs. This might break up the monotony of re-entering the castle and getting to the third floor after that star. After all, you need to find a level to bring you to the lobby anyway, right? To me, this seems like a good use for that alternate exit.