Basically there are limited things you can do to enemies in the game; restricted to jumping on them, using fireballs, or colliding with them while using invincibility. Now since there is no invincibility near the Bowser fight, and colliding with him (whether you jump on him or not) causes player damage, you're limited to Fireballs, and hitting the lever.
The only argument against this however is in the case of the "Imposters" or "Fake Bowsers", which will revert to their original forms once being hit by a fireball
and can be dealt with in their own respective manner (e.g. the Goomba can be squashed).
EDIT: @Michaellogg has pointed out that this last statement is actually incorrect; the "transformed sprite" is purely aesthetic, as hitting it with fireballs kills it outright.