Barrel Volcano

Hit at least 8 enemies. (For the 3rd room where you have a star, hold up in the corner until skeleton on the left walks toward you, then straight jump up.) If you get too many hits with the star, Geno will level up as well. He will appear first, and will look like Peach - so don't get confused. This will also cause Geno to learn the "Come Back" ability, so be aware of that. Additionally, Shocker may be on the 2nd page of Peach's specials now. If so, it will cost 2 FP per use (but you still need 8 FP stocked to use it).

In the Pyrosphere room, hug leftmost wall for good alignment, and then just jump across (you can't get hit).

In room with platforms, can jump on the two platforms and then make it to the left side with an angled jump (or simply jump straight ahead if you want an easier method). 

In room with save box, can open the 1st chest containing a Flower if you need it (this will typically happen anyways with normal movement).

In the next room, hug the rightmost wall and jump over the enemies to avoid getting an encounter, then proceed through the exit.

Be careful to avoid the Oerlikons dropped by the Corkpedite. The way in which the Oerlikons drop are random. If there are too many guarding the exit, you can reload the room and they will all be gone.

Since you've finished all your swap glitches, EXP route deviations from this point onward are not run-threatening. If you accidentally get into an encounter with a Stumpet, you cannot run. Have Mario untimed attack Mite (or Ice Bomb if you have a freebie), Geno Whirl (ジーノカッター) stumpet, Bowser defend, and Mario attack Mite after Valor Up. Do not play the Lucky Yoshi egg minigame. 

Talk to Hinopio from the upper-left to enter his cave, don't buy anything, then proceed across the bridge.

If you're struggling with the bridge, there is a slower but very easy method to get across:

(video credit: Sean)

If you want to go for TAS bridge, go to timestamp 1:34: