I recall having that song stuck in my head and coming up with one or two - but who knows what I'll remember now.

I mean, if the formula is to have a mono-syllable sound effect / onomatopoeic word that rhymes with a mono-syllable animal, then you could do the following:

Meow meow, I'm a cow! [this is the first alternate from the actual song]

Bark bark, I'm a shark!

Splish splish, I'm a fish!

Shake shake, I'm a snake!

Boom boom, I'm a poom! [puma]

Blam blam, I'm a ram! [a nice match for the original!]

Splat splat, I'm a cat! [ or bat or rat!]