The fourth bout with Fury Bowser can occur once you have collected at least 40 Cat Shines. As always, once you have the required number of Shines and Fury Bowser is out and about, go ahead and collect any of the Giga Bells in order to begin the boss fight. The fight will be mostly the same as the third bout, although Bowser will incorporate one new move into his arsenal, which you will see pretty much right away. After launching all of the blocks that fall down from the sky, Bowser will jump pretty far back and spit a few fireballs at you, while also simultaneously sending out some boulders.
The boulders he sends out are actually explosives and will detonate after a few seconds, but they will pulse red for a second or two before doing that, so that are easy to avoid. Also, simply running into them will cause Mario to kick them away, so there’s that, too. You should grab one of the blocks he shoots down from the sky and hold it until he starts his spinning charge attack, as you can simply block it, deal a little damage and also knock him onto his back. Once he is on his back, you can just perform a Ground Pound to deal more damage. Upon getting Fury Bowser’s health to around half, he will retreat until the next battle.