Why does this sentence take the instrumental instead of the genitive when it's negated? Is "Nie jestem małego dziecka" not also an acceptable translation?
No, it's not. You made a very common mistake of taking the 'negation rule' too far.
Many learners memorize the rule as "Negation -> Genitive". It's not a surprise, because Accusative is a very common case, and then negated Genitive (also common) is still Genitive. But there is no such rule. The rule is "Negated Accusative -> Genitive". Other cases stay unchanged when negated.