From the description it would seem like any Smash Hit would work, however it's either glitched or needs a better description as I have only found one way to make the counter in the Milestones screen go up.
First, you have to knock down a small enemy (throwing him will also work) and have the dizzy effect going on where birds fly over the enemy's head. When this happens, move over him, press down and mash the attack button repeatedly . Your 'Toad should start beating on the enemy if he has any amount of health left, then go to his smash hit attack (or go right to the smash hit attack if the enemy is low enough on health). Zitz will drill the enemy, Rash will claw the enemy's throat out, and Pimple's foot will become a weight that he crushes the enemy with.
This doesn't seem to work with large enemies as the 'Toads will boot them off screen. It's also hard to do it with weak enemies as they typically die in one hit before becoming dizzy.
The fastest way to grind this would be a spot with a few rats in level two (there are a few such areas) where you can easily take them out since there are only a few on the screen here at a time. What you can do is save the game, beat the rats in your area of choice, then reload the save and beat them over and over again.