Phase 1

There's not much strategy here. You can stomp multiple koopas if you're very accurate but it's difficult. You can grab koopas and throw them into others but it's generally not recommended, however, there's some sense in throwing a koopa if the target is invulnerable.

Phase 2

The snooze-fest of frightmare. This phase matters very little. You can get hit multiple times without it having an impact on your time.

Phase 3

This phase seems to rely almost entirely on RNG. Your goal is to get Bowser to jump onto the broken square twice. Bowser may jump, charge or breathe fire. Fortunately he seems to jump more often than not. I hoped there would be some way to manipulate the RNG but I've found no patterns (ex: the squares you run on, the distance between you and bowser when he performs an action, whether your character is jumping/running/standing).

Phase 4

This is where the real magic happens. Bowser will throw balls at you. The positioning is VERY important. The ideal positioning is slightly off-center to the left (he throws them from the left side). If they land in the center Bowser will push them too far away with his fire.

Bowser has 5 HP. Hopefully he will not throw more balls than you need or throw so few that you need an extra cycle to kill him. This mean you need 3 then 2 OR 2 than 3.

Immediately after the balls are activated by Bowser's fire, you should grab the ball that is furthest away from Bowser. Hopefully you are quick enough not be be hit by the fire the second time it comes over. Grab the activated balls and hit Bowser. It's POSSIBLE to hit him 4 times in one cycle but it's very hard (I've only done it once, 3 balls were on the left, 1 on the right). Each time Bowser gets hit, his "purple ball throw" animation gets reset. It's important that the last ball in the cycle doesn't reset this animation (AKA, wait for the wall to be out of his hand before he's hit).